Real periods of genus 2 curves over $\mathbb{Q}$

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Numbers

label

$\Omega_A$

169.a.169.1:

32.66703109050709611000590237

comment: Curve LMFDB 169.a.169.1 has isogeny class 169.a and model $y^2 + (x^3 + x + 1)y = x^5 + x^4$.

196.a.21952.1:

11.77714784061282915609229119

comment: Curve LMFDB 196.a.21952.1 has isogeny class 196.a and model $y^2 + (x^2 + x)y = x^6 + 3 x^5 + 6 x^4 + 7 x^3 + 6 x^2 + 3 x + 1$.

249.a.249.1:

25.78370337424983680582666323

comment: Curve LMFDB 249.a.249.1 has isogeny class 249.a and model $y^2 + (x^3 + 1)y = x^2 + x$.

249.a.6723.1:

25.78370337424983680582666323

comment: Curve LMFDB 249.a.6723.1 has isogeny class 249.a and model $y^2 + (x^3 + 1)y = -x^5 + x^3 + x^2 + 3 x + 2$.

256.a.512.1:

26.84182943692641124244186499

comment: Curve LMFDB 256.a.512.1 has isogeny class 256.a and model $y^2 + y = 2 x^5 - 3 x^4 + x^3 + x^2 - x$.

277.a.277.1:

32.20574862397758528503592224

comment: Curve LMFDB 277.a.277.1 has isogeny class 277.a and model $y^2 + (x^3 + x^2 + x + 1)y = -x^2 - x$.

277.a.277.2:

3.578416513775287253892880249

comment: Curve LMFDB 277.a.277.2 has isogeny class 277.a and model $y^2 + y = x^5 - 9 x^4 + 14 x^3 - 19 x^2 + 11 x - 6$.

294.a.294.1:

21.45153311338415631878371145

comment: Curve LMFDB 294.a.294.1 has isogeny class 294.a and model $y^2 + (x^3 + 1)y = x^4 + x^2$.

294.a.8232.1:

7.150511037794718772927903818

comment: Curve LMFDB 294.a.8232.1 has isogeny class 294.a and model $y^2 + (x^3 + 1)y = -2 x^4 + 4 x^2 - 9 x - 14$.

295.a.295.1:

29.25660030111076897552885992

comment: Curve LMFDB 295.a.295.1 has isogeny class 295.a and model $y^2 + (x^3 + 1)y = -x^2$.

295.a.295.2:

0.5970734755328728362352828556

comment: Curve LMFDB 295.a.295.2 has isogeny class 295.a and model $y^2 + (x^2 + x + 1)y = x^5 - 40 x^3 + 22 x^2 + 389 x - 608$.

324.a.648.1:

25.52176850296810024668026002

comment: Curve LMFDB 324.a.648.1 has isogeny class 324.a and model $y^2 + (x^3 + x + 1)y = x^5 + 2 x^4 + 2 x^3 + x^2$.

336.a.172032.1:

0.3560662442754820315373580806

comment: Curve LMFDB 336.a.172032.1 has isogeny class 336.a and model $y^2 + (x^3 + x)y = -x^6 + 15 x^4 - 75 x^2 - 56$.

349.a.349.1:

27.98848412463092306248410124

comment: Curve LMFDB 349.a.349.1 has isogeny class 349.a and model $y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x^2$.

353.a.353.1:

22.49549461731395943659449758

comment: Curve LMFDB 353.a.353.1 has isogeny class 353.a and model $y^2 + (x^3 + x + 1)y = x^2$.

360.a.6480.1:

24.16337901380689181478583546

comment: Curve LMFDB 360.a.6480.1 has isogeny class 360.a and model $y^2 + (x^3 + x)y = -3 x^4 + 7 x^2 - 5$.

363.a.11979.1:

18.97059599534884032426437150

comment: Curve LMFDB 363.a.11979.1 has isogeny class 363.a and model $y^2 + (x^2 + 1)y = x^5 + 2 x^3 + 4 x^2 + 2 x$.

363.a.43923.1:

3.794119199069768064852874300

comment: Curve LMFDB 363.a.43923.1 has isogeny class 363.a and model $y^2 + x^2 y = 11 x^5 - 13 x^4 - 7 x^3 + 10 x^2 + x - 2$.

388.a.776.1:

29.13550129458951056269076716

comment: Curve LMFDB 388.a.776.1 has isogeny class 388.a and model $y^2 + (x^3 + x + 1)y = -x^4 + 2 x^2 + x$.

389.a.389.1:

19.79862013008733330350209978

comment: Curve LMFDB 389.a.389.1 has isogeny class 389.a and model $y^2 + (x^3 + x)y = x^5 - 2 x^4 - 8 x^3 + 16 x + 7$.

389.a.389.2:

19.79862013008733330350209978

comment: Curve LMFDB 389.a.389.2 has isogeny class 389.a and model $y^2 + (x + 1)y = x^5 + 2 x^4 + 2 x^3 + x^2$.

394.a.394.1:

20.07827436393815644080807780

comment: Curve LMFDB 394.a.394.1 has isogeny class 394.a and model $y^2 + (x^3 + x)y = 2 x^5 + x^4 - 12 x^3 + 17 x - 9$.

394.a.3152.1:

20.07827436393815644080807780

comment: Curve LMFDB 394.a.3152.1 has isogeny class 394.a and model $y^2 + (x + 1)y = -x^5$.

400.a.409600.1:

7.977094942516564227224046949

comment: Curve LMFDB 400.a.409600.1 has isogeny class 400.a and model $y^2 = x^6 + 4 x^4 + 4 x^2 + 1$.

427.a.2989.1:

18.61317612582742400918645147

comment: Curve LMFDB 427.a.2989.1 has isogeny class 427.a and model $y^2 + (x^3 + 1)y = x^5 - x^4 - 5 x^3 + 4 x^2 + 4 x - 4$.

448.a.448.1:

7.792788965311644825093853187

comment: Curve LMFDB 448.a.448.1 has isogeny class 448.a and model $y^2 + (x^3 + x)y = x^4 - 7$.

448.a.448.2:

31.17115586124657930037541275

comment: Curve LMFDB 448.a.448.2 has isogeny class 448.a and model $y^2 + (x^3 + x)y = -2 x^4 + 7$.

450.a.2700.1:

18.77899571854043724939357821

comment: Curve LMFDB 450.a.2700.1 has isogeny class 450.a and model $y^2 + (x^3 + 1)y = x^5 + 3 x^4 + 3 x^3 + 3 x^2 + x$.

450.a.36450.1:

18.77899571854043724939357821

comment: Curve LMFDB 450.a.36450.1 has isogeny class 450.a and model $y^2 + (x^3 + 1)y = x^5 - 4 x^4 - 9 x^3 + 28 x^2 - 6 x - 16$.

461.a.461.1:

12.04843489716017965610631486

comment: Curve LMFDB 461.a.461.1 has isogeny class 461.a and model $y^2 + x^3 y = x^5 - 3 x^3 + 3 x - 2$.

461.a.461.2:

0.2458864264726567276756390789

comment: Curve LMFDB 461.a.461.2 has isogeny class 461.a and model $y^2 + y = x^5 - x^4 - 39 x^3 + 10 x^2 + 272 x - 306$.

464.a.464.1:

14.42143083210724472918897778

comment: Curve LMFDB 464.a.464.1 has isogeny class 464.a and model $y^2 + (x + 1)y = -x^6 - 2 x^5 - 2 x^4 - x^3$.

464.a.29696.1:

14.42143083210724472918897778

comment: Curve LMFDB 464.a.29696.1 has isogeny class 464.a and model $y^2 + (x + 1)y = 8 x^5 + 3 x^4 - 4 x^3 - 2 x^2$.

464.a.29696.2:

1.802678854013405591148622223

comment: Curve LMFDB 464.a.29696.2 has isogeny class 464.a and model $y^2 + xy = 4 x^5 + 33 x^4 + 72 x^3 + 16 x^2 + x$.

472.a.944.1:

29.11327330457606774144841846

comment: Curve LMFDB 472.a.944.1 has isogeny class 472.a and model $y^2 + (x^2 + 1)y = x^5 - x^4 - 2 x^3 + x$.

472.a.60416.1:

7.278318326144016935362104615

comment: Curve LMFDB 472.a.60416.1 has isogeny class 472.a and model $y^2 + (x + 1)y = 8 x^5 + 5 x^4 + 4 x^3 + 2 x^2$.

476.a.952.1:

26.72233910632963251214860252

comment: Curve LMFDB 476.a.952.1 has isogeny class 476.a and model $y^2 + (x^3 + 1)y = -5 x^4 + 7 x^3 + 25 x^2 - 75 x + 54$.

484.a.1936.1:

15.31896832265624037237930824

comment: Curve LMFDB 484.a.1936.1 has isogeny class 484.a and model $y^2 + y = x^6 + 2 x^4 + x^2$.

504.a.27216.1:

7.782699444387423938615175676

comment: Curve LMFDB 504.a.27216.1 has isogeny class 504.a and model $y^2 + (x^3 + x)y = 3 x^4 + 15 x^2 + 21$.

523.a.523.1:

24.81990412677571630041171344

comment: Curve LMFDB 523.a.523.1 has isogeny class 523.a and model $y^2 + (x + 1)y = x^5 - x^4 - x^3$.

523.a.523.2:

0.9927961650710286520164685376

comment: Curve LMFDB 523.a.523.2 has isogeny class 523.a and model $y^2 + xy = x^5 - 31 x^4 - 110 x^3 + 21 x^2 - x$.

529.a.529.1:

30.06025585746256142607729105

comment: Curve LMFDB 529.a.529.1 has isogeny class 529.a and model $y^2 + (x^3 + x + 1)y = -x^5$.

555.a.8325.1:

25.69247202971451430840301627

comment: Curve LMFDB 555.a.8325.1 has isogeny class 555.a and model $y^2 + (x + 1)y = 3 x^5 - 2 x^4 - 4 x^3 + x^2 + x$.

574.a.293888.1:

11.54635017206091601529730074

comment: Curve LMFDB 574.a.293888.1 has isogeny class 574.a and model $y^2 + (x^2 + x)y = x^5 - x^4 - 3 x^2 + x + 1$.

576.a.576.1:

22.39625216818580607766497752

comment: Curve LMFDB 576.a.576.1 has isogeny class 576.a and model $y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x$.

576.b.147456.1:

9.301119475821611039064116900

comment: Curve LMFDB 576.b.147456.1 has isogeny class 576.b and model $y^2 = x^6 + 2 x^4 + 2 x^2 + 1$.

578.a.2312.1:

13.91029942184442728940055335

comment: Curve LMFDB 578.a.2312.1 has isogeny class 578.a and model $y^2 + (x^2 + x)y = x^5 - 2 x^4 + 2 x^3 - 2 x^2 + x$.

587.a.587.1:

29.51096406109354582967022954

comment: Curve LMFDB 587.a.587.1 has isogeny class 587.a and model $y^2 + (x^3 + x + 1)y = -x^2 - x$.

588.a.18816.1:

20.65814959093327320416213484

comment: Curve LMFDB 588.a.18816.1 has isogeny class 588.a and model $y^2 + (x^3 + 1)y = x^5 + x^4 + 5 x^2 + 12 x + 8$.

597.a.597.1:

14.41161687922246816026056494

comment: Curve LMFDB 597.a.597.1 has isogeny class 597.a and model $y^2 + y = x^5 + 2 x^4 + 3 x^3 + 2 x^2 + x$.

600.a.18000.1:

18.93431868106961183611104952

comment: Curve LMFDB 600.a.18000.1 has isogeny class 600.a and model $y^2 + xy = 10 x^5 - 18 x^4 + 8 x^3 + x^2 - x$.

600.a.96000.1:

9.467159340534805918055524762

comment: Curve LMFDB 600.a.96000.1 has isogeny class 600.a and model $y^2 + (x + 1)y = 4 x^5 + 5 x^4 + 3 x^3 + 2 x^2$.

600.b.30000.1:

8.316290505407428834943190971

comment: Curve LMFDB 600.b.30000.1 has isogeny class 600.b and model $y^2 + (x^3 + x)y = x^4 + x^2 - 3$.

600.b.450000.1:

8.316290505407428834943190971

comment: Curve LMFDB 600.b.450000.1 has isogeny class 600.b and model $y^2 + (x^3 + x)y = -5 x^4 + 25 x^2 - 45$.

603.a.603.1:

26.91001556363177690790850827

comment: Curve LMFDB 603.a.603.1 has isogeny class 603.a and model $y^2 + (x^2 + 1)y = x^5 + 8 x^4 + 4 x^3 + 4 x^2 + 2 x$.

603.a.603.2:

26.91001556363177690790850827

comment: Curve LMFDB 603.a.603.2 has isogeny class 603.a and model $y^2 + (x^2 + 1)y = x^5 - x^3 + x$.

604.a.9664.1:

0.2917880340909438023610651534

comment: Curve LMFDB 604.a.9664.1 has isogeny class 604.a and model $y^2 + (x^2 + x + 1)y = 4 x^5 + 9 x^4 + 48 x^3 - 4 x^2 - 53 x - 21$.

604.a.9664.2:

23.63483076136644799124627742

comment: Curve LMFDB 604.a.9664.2 has isogeny class 604.a and model $y^2 + (x^3 + 1)y = -x^4 + x^3 + x^2 - x$.

630.a.34020.1:

19.47088912974860787735976054

comment: Curve LMFDB 630.a.34020.1 has isogeny class 630.a and model $y^2 + (x^2 + x)y = 3 x^5 + 10 x^4 - 23 x^2 - 6 x + 15$.

640.a.81920.1:

7.405674177128772628232812421

comment: Curve LMFDB 640.a.81920.1 has isogeny class 640.a and model $y^2 + x^3 y = 3 x^4 + 13 x^2 + 20$.

640.a.81920.2:

7.405674177128772628232812421

comment: Curve LMFDB 640.a.81920.2 has isogeny class 640.a and model $y^2 + x^3 y = -3 x^4 + 13 x^2 - 20$.

644.a.2576.1:

3.928431134109850626488427461

comment: Curve LMFDB 644.a.2576.1 has isogeny class 644.a and model $y^2 + (x^2 + x)y = -5 x^6 + 11 x^5 - 20 x^4 + 20 x^3 - 20 x^2 + 11 x - 5$.

644.a.659456.1:

0.8729846964688556947752061026

comment: Curve LMFDB 644.a.659456.1 has isogeny class 644.a and model $y^2 + (x^2 + x)y = -3 x^6 - 13 x^5 + 4 x^4 + 51 x^3 + 4 x^2 - 13 x - 3$.

644.b.14812.1:

15.43510719899337302636569541

comment: Curve LMFDB 644.b.14812.1 has isogeny class 644.b and model $y^2 + (x^3 + 1)y = x^5 - x^4 - 4 x^3 + 5 x^2 - x - 1$.

672.a.172032.1:

1.113349301814943830917553304

comment: Curve LMFDB 672.a.172032.1 has isogeny class 672.a and model $y^2 + (x^3 + x)y = -x^6 - 16 x^4 - 75 x^2 + 56$.

676.a.5408.1:

20.16977983522340726059302496

comment: Curve LMFDB 676.a.5408.1 has isogeny class 676.a and model $y^2 + (x^3 + x^2 + x)y = x^3 + 3 x^2 + 3 x + 1$.

676.a.562432.1:

6.723259945074469086864341653

comment: Curve LMFDB 676.a.562432.1 has isogeny class 676.a and model $y^2 + (x^3 + 1)y = 2 x^5 + 2 x^4 + 4 x^3 + 2 x^2 + 2 x$.

676.b.17576.1:

7.177120650242340022568216320

comment: Curve LMFDB 676.b.17576.1 has isogeny class 676.b and model $y^2 + (x^2 + x)y = -x^6 + 3 x^5 - 6 x^4 + 6 x^3 - 6 x^2 + 3 x - 1$.

686.a.686.1:

11.49165478948144714805112930

comment: Curve LMFDB 686.a.686.1 has isogeny class 686.a and model $y^2 + (x^2 + x)y = x^5 + x^4 + 2 x^3 + x^2 + x$.

688.a.2752.1:

25.70729814990717701123340693

comment: Curve LMFDB 688.a.2752.1 has isogeny class 688.a and model $y^2 + y = 2 x^5 - 5 x^4 + 4 x^3 - x$.

688.a.704512.1:

6.426824537476794252808351734

comment: Curve LMFDB 688.a.704512.1 has isogeny class 688.a and model $y^2 = 2 x^5 + 4 x^3 + x^2 + 2 x + 1$.

688.a.704512.2:

6.426824537476794252808351734

comment: Curve LMFDB 688.a.704512.2 has isogeny class 688.a and model $y^2 = 2 x^5 - 7 x^4 - 8 x^3 + 2 x^2 + 4 x + 1$.

691.a.691.1:

18.81256891262843151418253621

comment: Curve LMFDB 691.a.691.1 has isogeny class 691.a and model $y^2 + (x + 1)y = x^5 - x^3 - x^2$.

704.a.45056.1:

11.97602710419342856089487448

comment: Curve LMFDB 704.a.45056.1 has isogeny class 704.a and model $y^2 + y = 4 x^5 + 4 x^4 - x^3 - 2 x^2$.

708.a.2832.1:

16.26718131545465430994887232

comment: Curve LMFDB 708.a.2832.1 has isogeny class 708.a and model $y^2 + (x^2 + x + 1)y = x^5$.

708.a.19116.1:

16.26718131545465430994887232

comment: Curve LMFDB 708.a.19116.1 has isogeny class 708.a and model $y^2 + (x^3 + 1)y = -x^5 + 4 x^2 + 4 x - 1$.

708.a.181248.1:

0.3253436263090930861989774465

comment: Curve LMFDB 708.a.181248.1 has isogeny class 708.a and model $y^2 + (x^3 + 1)y = -x^6 - 4 x^5 + 9 x^4 + 48 x^3 - 41 x^2 - 98 x - 36$.

709.a.709.1:

18.36116163245614548037310111

comment: Curve LMFDB 709.a.709.1 has isogeny class 709.a and model $y^2 + xy = x^5 - 2 x^2 + x$.

713.a.713.1:

27.95788881235783899596312622

comment: Curve LMFDB 713.a.713.1 has isogeny class 713.a and model $y^2 + (x^3 + x + 1)y = -x^5 - x$.

713.b.713.1:

23.14988107670389907883138784

comment: Curve LMFDB 713.b.713.1 has isogeny class 713.b and model $y^2 + (x^3 + x + 1)y = -x^4$.

720.a.6480.1:

9.444268302255141344599789586

comment: Curve LMFDB 720.a.6480.1 has isogeny class 720.a and model $y^2 + (x^3 + x)y = 2 x^4 + 7 x^2 + 5$.

720.b.116640.1:

14.45705774132695543640100863

comment: Curve LMFDB 720.b.116640.1 has isogeny class 720.b and model $y^2 + (x^3 + x)y = -6 x^4 + 39 x^2 - 90$.

726.a.1452.1:

15.12408611092730690139582852

comment: Curve LMFDB 726.a.1452.1 has isogeny class 726.a and model $y^2 + (x^2 + 1)y = 2 x^5 + 2 x^4 + 6 x^3 - 2 x^2 - x$.

731.a.12427.1:

14.92677943436207031431594936

comment: Curve LMFDB 731.a.12427.1 has isogeny class 731.a and model $y^2 + (x^3 + x^2)y = x^5 + 2 x^4 - x - 3$.

741.a.28899.1:

18.75684256809198078031270145

comment: Curve LMFDB 741.a.28899.1 has isogeny class 741.a and model $y^2 + (x + 1)y = -3 x^5 - x^4 + 2 x^2 + x$.

743.a.743.1:

28.76539061736102992978107429

comment: Curve LMFDB 743.a.743.1 has isogeny class 743.a and model $y^2 + (x^3 + x + 1)y = -x^4 + x^2$.

745.a.745.1:

24.57283976052654794911342517

comment: Curve LMFDB 745.a.745.1 has isogeny class 745.a and model $y^2 + (x^3 + x + 1)y = -x$.

762.a.3048.1:

16.73344918309254302919366102

comment: Curve LMFDB 762.a.3048.1 has isogeny class 762.a and model $y^2 + (x^3 + x^2 + x)y = x^2 + x + 1$.

762.a.82296.1:

16.73344918309254302919366102

comment: Curve LMFDB 762.a.82296.1 has isogeny class 762.a and model $y^2 + (x^2 + x)y = x^5 - 8 x^4 + 14 x^3 + 2 x^2 - x$.

763.a.763.1:

30.48575004522535920085251501

comment: Curve LMFDB 763.a.763.1 has isogeny class 763.a and model $y^2 + (x^3 + x)y = -2 x^4 + 2 x^2 - x$.

768.a.1536.1:

25.14674941491256855456653960

comment: Curve LMFDB 768.a.1536.1 has isogeny class 768.a and model $y^2 + y = 2 x^5 - x^4 - 3 x^3 + x$.

768.a.4608.1:

12.57337470745628427728326980

comment: Curve LMFDB 768.a.4608.1 has isogeny class 768.a and model $y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x^2 - x - 1$.

784.a.1568.1:

20.79335121010696255321919782

comment: Curve LMFDB 784.a.1568.1 has isogeny class 784.a and model $y^2 + (x^3 + x)y = -2 x^4 + 3 x^2 - 2$.

784.a.43904.1:

6.931117070035654184406399276

comment: Curve LMFDB 784.a.43904.1 has isogeny class 784.a and model $y^2 + (x^3 + x)y = 4 x^4 + 27 x^2 + 56$.

784.b.12544.1:

11.27010029351507222440474365

comment: Curve LMFDB 784.b.12544.1 has isogeny class 784.b and model $y^2 + (x^3 + x)y = -1$.

784.b.25088.1:

0.6261166829730595680224857584

comment: Curve LMFDB 784.b.25088.1 has isogeny class 784.b and model $y^2 + (x^2 + 1)y = -x^6 - 3 x^5 + 7 x^4 + 2 x^3 - 49 x^2 + 41 x - 9$.

784.b.76832.1:

3.756700097838357408134914550

comment: Curve LMFDB 784.b.76832.1 has isogeny class 784.b and model $y^2 + (x + 1)y = -x^6 + 4 x^5 - 4 x^4 - 2 x^3 + 10 x - 9$.

784.c.614656.1:

5.731485289101440494387615136

comment: Curve LMFDB 784.c.614656.1 has isogeny class 784.c and model $y^2 = x^5 - 4 x^4 - 13 x^3 - 9 x^2 - x$.

797.a.797.1:

17.44098947627626520565187602

comment: Curve LMFDB 797.a.797.1 has isogeny class 797.a and model $y^2 + y = x^5 - x^4 + x^3$.

800.a.1600.1:

16.77015076139292791148777638

comment: Curve LMFDB 800.a.1600.1 has isogeny class 800.a and model $y^2 + (x^3 + x^2 + x + 1)y = -x^4 - x^2$.

800.a.8000.1:

5.590050253797642637162592129

comment: Curve LMFDB 800.a.8000.1 has isogeny class 800.a and model $y^2 + (x^3 + x^2 + x + 1)y = -x^6 + 2 x^4 + 4 x^3 + 2 x^2 - 1$.

800.a.409600.1:

16.77015076139292791148777638

comment: Curve LMFDB 800.a.409600.1 has isogeny class 800.a and model $y^2 = x^6 - 2 x^2 + 1$.

807.a.2421.1:

9.761140262957613257420905360

comment: Curve LMFDB 807.a.2421.1 has isogeny class 807.a and model $y^2 + (x^3 + x)y = x^5 - 2 x^3 - x^2 + 2 x - 1$.

810.a.196830.1:

0.3289823157626603735056020514

comment: Curve LMFDB 810.a.196830.1 has isogeny class 810.a and model $y^2 + (x + 1)y = x^5 + 15 x^4 + 20 x^3 - 297 x^2 + 94 x - 8$.

816.a.13872.1:

22.16669692787501924452403723

comment: Curve LMFDB 816.a.13872.1 has isogeny class 816.a and model $y^2 + (x^3 + x^2)y = -2 x^4 + 6 x^2 - 8 x + 3$.

816.a.39168.1:

22.16669692787501924452403723

comment: Curve LMFDB 816.a.39168.1 has isogeny class 816.a and model $y^2 + (x^2 + 1)y = 3 x^5 - 4 x^3 - x^2 + x$.

816.b.52224.1:

2.423742077271449078203055296

comment: Curve LMFDB 816.b.52224.1 has isogeny class 816.b and model $y^2 + (x^3 + x)y = -x^6 - 12 x^4 - 27 x^2 - 17$.

826.a.11564.1:

13.17448343507314962222933299

comment: Curve LMFDB 826.a.11564.1 has isogeny class 826.a and model $y^2 + (x^2 + x)y = x^5 + x^4 + 3 x^3 - 4 x^2 - 4 x + 3$.

830.a.6640.1:

23.47491713469872304349835866

comment: Curve LMFDB 830.a.6640.1 has isogeny class 830.a and model $y^2 + (x^3 + 1)y = -x^5 + x^4 - 2 x^2 + x + 1$.

830.a.830000.1:

5.868729283674680760874589666

comment: Curve LMFDB 830.a.830000.1 has isogeny class 830.a and model $y^2 + (x^2 + x)y = x^5 - 2 x^4 + 16 x^3 + 8 x^2 + x$.

832.a.832.1:

21.14821534737590344234962396

comment: Curve LMFDB 832.a.832.1 has isogeny class 832.a and model $y^2 + (x^3 + x)y = x^5 - x^3 + x^2 + 2 x - 1$.

834.a.1668.1:

11.76351618255668262122669454

comment: Curve LMFDB 834.a.1668.1 has isogeny class 834.a and model $y^2 + (x^3 + 1)y = -x^2 + x - 1$.

841.a.841.1:

14.28455671925849588123892420

comment: Curve LMFDB 841.a.841.1 has isogeny class 841.a and model $y^2 + (x^3 + x^2 + x)y = x^4 + x^3 + 3 x^2 + x + 2$.

847.a.847.1:

20.30596137898254740951507105

comment: Curve LMFDB 847.a.847.1 has isogeny class 847.a and model $y^2 + (x^3 + x^2 + x + 1)y = x^4 + x^3 + x^2$.

847.b.9317.1:

16.82727124881458838085059429

comment: Curve LMFDB 847.b.9317.1 has isogeny class 847.b and model $y^2 + (x^2 + 1)y = x^5 + 2 x^4 - 3 x^3 + 2 x^2 - x$.

847.c.9317.1:

9.983399950485201184125312381

comment: Curve LMFDB 847.c.9317.1 has isogeny class 847.c and model $y^2 + (x^3 + x^2)y = x^4 + x^3 - x - 2$.

847.d.847.1:

1.179534640077608274864061558

comment: Curve LMFDB 847.d.847.1 has isogeny class 847.d and model $y^2 + (x^3 + x^2 + x + 1)y = -12 x^6 - 15 x^5 + 9 x^4 + 31 x^3 + 9 x^2 - 15 x - 12$.

847.d.456533.1:

9.829455333980068957200512987

comment: Curve LMFDB 847.d.456533.1 has isogeny class 847.d and model $y^2 + y = -x^6 - 9 x^5 - 22 x^4 + 3 x^3 + 37 x^2 - 24 x + 4$.

856.a.1712.1:

22.65384566628467976754192158

comment: Curve LMFDB 856.a.1712.1 has isogeny class 856.a and model $y^2 + (x^3 + x)y = -x^4 - x^3 + x$.

862.a.862.1:

23.92660509771110912862980413

comment: Curve LMFDB 862.a.862.1 has isogeny class 862.a and model $y^2 + (x^3 + 1)y = x^5 - 2 x^4 - 7 x^3 + 7 x^2 + 2 x + 5$.

862.a.6896.1:

23.92660509771110912862980413

comment: Curve LMFDB 862.a.6896.1 has isogeny class 862.a and model $y^2 + (x^2 + x)y = 4 x^5 + 6 x^4 - 3 x^2 - x$.

862.b.862.1:

27.48899057752936272094588589

comment: Curve LMFDB 862.b.862.1 has isogeny class 862.b and model $y^2 + (x^3 + x)y = -2 x^4 + 3 x^2 - x - 1$.

864.a.1728.1:

18.14296590467000967231465706

comment: Curve LMFDB 864.a.1728.1 has isogeny class 864.a and model $y^2 + (x^3 + x^2 + x + 1)y = x^4 + x^2$.

864.a.221184.1:

18.14296590467000967231465706

comment: Curve LMFDB 864.a.221184.1 has isogeny class 864.a and model $y^2 + x^3 y = x^5 - 4 x^4 - 6 x^3 + 33 x^2 - 36 x + 12$.

864.a.442368.1:

9.071482952335004836157328533

comment: Curve LMFDB 864.a.442368.1 has isogeny class 864.a and model $y^2 = x^6 - 4 x^4 + 6 x^2 - 3$.

880.a.225280.1:

1.515081981797020835404899726

comment: Curve LMFDB 880.a.225280.1 has isogeny class 880.a and model $y^2 = x^5 + 13 x^4 + 55 x^3 + 76 x^2 - 44$.

882.a.63504.1:

12.54262310956065397709650759

comment: Curve LMFDB 882.a.63504.1 has isogeny class 882.a and model $y^2 + (x^2 + x)y = x^5 + x^4 + x^3 + 3 x^2 + 3 x + 1$.

882.a.302526.1:

12.54262310956065397709650759

comment: Curve LMFDB 882.a.302526.1 has isogeny class 882.a and model $y^2 + (x^3 + 1)y = x^5 - 2 x^4 - 5 x^3 + 11 x^2 - 12 x + 5$.

886.a.3544.1:

24.08547171248876178872970699

comment: Curve LMFDB 886.a.3544.1 has isogeny class 886.a and model $y^2 + (x^3 + x)y = -x^4 - x + 1$.

893.a.893.1:

23.40243481037167042315048833

comment: Curve LMFDB 893.a.893.1 has isogeny class 893.a and model $y^2 + (x^3 + x + 1)y = -x^4 - x^2$.

909.a.909.1:

21.80554843268429195244721538

comment: Curve LMFDB 909.a.909.1 has isogeny class 909.a and model $y^2 + (x^3 + x)y = -x^4 + x^2 - x$.

909.a.8181.1:

21.80554843268429195244721538

comment: Curve LMFDB 909.a.8181.1 has isogeny class 909.a and model $y^2 + xy = 3 x^5 - 7 x^4 + x^3 + 6 x^2 - 3 x$.

925.a.925.1:

20.87893399657743451578615273

comment: Curve LMFDB 925.a.925.1 has isogeny class 925.a and model $y^2 + (x + 1)y = -x^5 + 2 x^4 - x^3 - x^2$.

925.a.23125.1:

20.87893399657743451578615273

comment: Curve LMFDB 925.a.23125.1 has isogeny class 925.a and model $y^2 + xy = 5 x^5 + x^4 - 19 x^3 + 18 x^2 - 5 x$.

930.a.930.1:

24.84648867296860639657622856

comment: Curve LMFDB 930.a.930.1 has isogeny class 930.a and model $y^2 + (x^2 + x)y = -x^5 - 7 x^4 + 37 x^2 - 45 x + 15$.

932.a.3728.1:

25.16836447211651148952328278

comment: Curve LMFDB 932.a.3728.1 has isogeny class 932.a and model $y^2 + y = x^6 - 2 x^5 + x^4 + x^2 - x$.

936.a.1872.1:

7.131061142300018139648377262

comment: Curve LMFDB 936.a.1872.1 has isogeny class 936.a and model $y^2 + (x^3 + x)y = -x^6 - 9 x^4 - 32 x^2 - 39$.

953.a.953.1:

24.88668171534092899854470851

comment: Curve LMFDB 953.a.953.1 has isogeny class 953.a and model $y^2 + (x^3 + x + 1)y = x^3 + x^2$.

960.a.245760.1:

6.402317452558049353443606128

comment: Curve LMFDB 960.a.245760.1 has isogeny class 960.a and model $y^2 = 2 x^5 + x^4 + 4 x^3 + x^2 + 2 x$.

960.a.368640.1:

6.402317452558049353443606128

comment: Curve LMFDB 960.a.368640.1 has isogeny class 960.a and model $y^2 = x^5 + 13 x^4 + 44 x^3 + 13 x^2 + x$.

960.a.983040.1:

6.402317452558049353443606128

comment: Curve LMFDB 960.a.983040.1 has isogeny class 960.a and model $y^2 = x^5 - 2 x^4 - x^3 - 2 x^2 + x$.

961.a.961.1:

0.2246438619380203930566648340

comment: Curve LMFDB 961.a.961.1 has isogeny class 961.a and model $y^2 + (x^3 + x + 1)y = -x^6 - x^5 - 7 x^4 + 74 x^3 - 145 x^2 + 99 x - 33$.

961.a.961.2:

5.616096548450509826416620852

comment: Curve LMFDB 961.a.961.2 has isogeny class 961.a and model $y^2 + (x^3 + x + 1)y = -x^6 + 2 x^5 - 8 x^4 + 12 x^3 - 18 x^2 + 12 x - 7$.

961.a.961.3:

11.23219309690101965283324170

comment: Curve LMFDB 961.a.961.3 has isogeny class 961.a and model $y^2 + (x^3 + x + 1)y = x^5 + x^4 + x^3 - x - 1$.

961.a.923521.1:

2.246438619380203930566648340

comment: Curve LMFDB 961.a.923521.1 has isogeny class 961.a and model $y^2 + (x^3 + x^2 + 1)y = -5 x^4 + 4 x^3 + 3 x^2 - 2 x - 3$.

966.a.834624.1:

9.526770801078762144556769365

comment: Curve LMFDB 966.a.834624.1 has isogeny class 966.a and model $y^2 + (x^2 + x)y = x^5 - x^4 + x^3 + x^2 - x + 1$.

968.a.1936.1:

26.98675025097742069166883368

comment: Curve LMFDB 968.a.1936.1 has isogeny class 968.a and model $y^2 + y = x^6 - x^4$.

968.a.234256.1:

5.397350050195484138333766737

comment: Curve LMFDB 968.a.234256.1 has isogeny class 968.a and model $y^2 + x^3 y = 6 x^4 + 47 x^2 + 121$.

970.a.1940.1:

17.37577237002508223734326945

comment: Curve LMFDB 970.a.1940.1 has isogeny class 970.a and model $y^2 + (x + 1)y = x^5 + x^4 + x^3 + x^2$.

971.a.971.1:

29.64711080112498856304281948

comment: Curve LMFDB 971.a.971.1 has isogeny class 971.a and model $y^2 + y = x^5 - 2 x^3 + x$.

975.a.63375.1:

14.35629040776369988366316747

comment: Curve LMFDB 975.a.63375.1 has isogeny class 975.a and model $y^2 + (x^3 + 1)y = -x^5 + x^3 + 2 x^2 + x - 1$.

976.a.999424.1:

12.90036505815206467681008301

comment: Curve LMFDB 976.a.999424.1 has isogeny class 976.a and model $y^2 + (x + 1)y = x^6 - 2 x^5 + 2 x^3 - x^2$.

980.a.7840.1:

14.03151913657681812783628370

comment: Curve LMFDB 980.a.7840.1 has isogeny class 980.a and model $y^2 + (x^2 + x + 1)y = -x^6 + 3 x^5 - 3 x^4 - x$.

980.a.878080.1:

4.677173045525606042612094566

comment: Curve LMFDB 980.a.878080.1 has isogeny class 980.a and model $y^2 + (x^3 + 1)y = -x^6 + x^5 - 4 x^4 + 2 x^3 - 4 x^2 + x - 1$.

990.a.8910.1:

6.174936843698375280393216721

comment: Curve LMFDB 990.a.8910.1 has isogeny class 990.a and model $y^2 + (x^2 + x)y = 3 x^5 + 4 x^4 + 7 x^3 + 4 x^2 + 3 x$.

990.a.240570.1:

3.087468421849187640196608360

comment: Curve LMFDB 990.a.240570.1 has isogeny class 990.a and model $y^2 + (x^2 + x)y = 3 x^5 + 28 x^4 + 72 x^3 + 28 x^2 + 3 x$.

997.a.997.1:

21.58962105557654397482680362

comment: Curve LMFDB 997.a.997.1 has isogeny class 997.a and model $y^2 + xy = x^5 - 8 x^4 + 16 x^3 - x$.

997.a.997.2:

21.58962105557654397482680362

comment: Curve LMFDB 997.a.997.2 has isogeny class 997.a and model $y^2 + (x + 1)y = x^5 + x^4$.

997.b.997.1:

19.93284336545544441524084864

comment: Curve LMFDB 997.b.997.1 has isogeny class 997.b and model $y^2 + y = x^5 - 2 x^4 + 2 x^3 - x^2$.

Definition

Let $C/\mathbb{Q}$ be a genus $2$ curve with Jacobian $A=\operatorname{Jac}(C)$ and period lattice $\Lambda$. This table gives $\Omega_A=\#\pi_0(A(\mathbb{R}))\operatorname{covol}(\Lambda\cap\mathbb{R}^2)$, with covolume in $\mathbb{R}^2$ [1].

Parameters

label

— LMFDB curve label

Formulas

(1)

The Birch and Swinnerton-Dyer conjecture predicts $L^{(r)}(A,1)/r! = \Omega_A R_A |\operatorname{Sha}(A)| \prod_p c_p / |A(\mathbb{Q})_{\mathrm{tors}}|^2$, where $r$ is the analytic rank, $R_A$ is the regulator, $c_p$ is the Tamagawa number at $p$, and $A(\mathbb{Q})_{\mathrm{tors}}$ is the torsion subgroup [3].

Comments

(2)

The label is the LMFDB label for the genus $2$ curve [2]. Each entry comment gives the isogeny class of the Jacobian and the model $y^2+h(x)y=f(x)$.

(3)

If $\omega_1,\omega_2$ is the basis of the global sections of the canonical sheaf on a regular model of $C$ over $\mathbb{Z}$, then $\Lambda$ is generated by the vectors $(\int_\gamma\omega_1,\int_\gamma\omega_2)$ for $\gamma\in H_1(C(\mathbb{C}),\mathbb{Z})$.

(4)

The connected-component factor is part of $\Omega_A$ in this table. Some authors omit this factor from a period covolume, so the convention should be checked when comparing with another source.

(5)

The real period is an invariant of the Jacobian and not of its isogeny class: 277.a.277.1 and 277.a.277.2 lie in one class and have different periods, while 249.a.249.1 and 249.a.6723.1 have the same period.

Links

[1]

LMFDB knowl: real period of a genus 2 curve

[2]

LMFDB: genus 2 curves over $\mathbb{Q}$

[3]

LMFDB knowl: BSD invariants for genus 2 curves

[4]

LMFDB JSON data: genus 2 curve 169.a.169.1

Similar tables

Real periods of elliptic curves over $\mathbb{Q}$ of rank $1$ — stores the genus $1$ real period for elliptic curves of rank $1$

Real periods of elliptic curves over $\mathbb{Q}$ of rank $2$ — stores the genus $1$ real period for elliptic curves of rank $2$

Real periods of elliptic curves over $\mathbb{Q}$ of rank $3$ — stores the genus $1$ real period for elliptic curves of rank $3$

Special $L$-values of genus $2$ curves over $\mathbb{Q}$ — stores the $L$-function factor that appears with the real period in the BSD formula for the same genus $2$ Jacobians

Equations satisfied by Igusa invariants of split Jacobians — stores equations for genus $2$ curves whose Jacobians split

Data properties

Entries are of type: real number

How well the digits are known: heuristic

Table is complete: no (it holds every LMFDB genus $2$ curve with conductor $N\leq1000$ whose real period is recorded in the LMFDB genus $2$ curve data)

How they were obtained:

From the frozen $159$ LMFDB genus $2$ curve rows with conductor $N\leq1000$, fetched from g2c_curves, the generator transcribes the real_period decimal string and stores the first $28$ significant digits, truncated and not rounded, one digit fewer than the source's $97$-bit precision supports [4].

Before the draft was created, PARI's lfungenus2 computation matched the LMFDB leading_coeff values on the $41$ odd-conductor isogeny classes in the range, and the BSD quotient in (1) reproduced the $56$ odd-conductor real periods to the precision supported by the source fields. The even-conductor rows are not certified here by an independent period computation; the table records the LMFDB values and is therefore marked heuristic.