Kernel polynomials of rational isogenies of elliptic curves over $\mathbb{Q}$

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Polynomials

$N$

$c_4$, $c_6$

$\ell$

$c'_4$, $c'_6$

$K_\varphi(x)$

496, 20008

375376, 229985128:

x^2 - 21*x + 80

comment: Cremona labels 11a1 to 11a2.

496, 20008

16, -152:

x^2 + x - 29/5

comment: Cremona labels 11a1 to 11a3.

375376, 229985128

496, 20008:

x^2 + 101*x + 12751/5

comment: Cremona labels 11a2 to 11a1.

16, -152

496, 20008:

x^2 - x

comment: Cremona labels 11a3 to 11a1.

129, -2241

10209, 1131759:

x^3 - 3*x^2 - x + 3

comment: Cremona labels 26b1 to 26b2.

10209, 1131759

129, -2241:

x^3 + 15*x^2 - 94*x - 7237/7

comment: Cremona labels 26b2 to 26b1.

1, -1025

3361, 215695:

x^2 - 1

comment: Cremona labels 38b1 to 38b2.

3361, 215695

1, -1025:

x^2 + 5*x - 59/5

comment: Cremona labels 38b2 to 38b1.

105, 1323

5145, -453789:

x^3 + x^2 - 2*x - 1

comment: Cremona labels 49a1 to 49a3.

1785, 75411

87465, -25865973:

x^3 + 15*x^2 + 68*x + 97

comment: Cremona labels 49a2 to 49a4.

5145, -453789

105, 1323:

x^3 - 13*x^2 - 58*x + 503

comment: Cremona labels 49a3 to 49a1.

87465, -25865973

1785, 75411:

x^3 - 111*x^2 + 3764*x - 40363

comment: Cremona labels 49a4 to 49a2.

25, 1475

3625, -263125:

x^2 + x - 1

comment: Cremona labels 50a1 to 50a3.

6025, 467675

-26375, 1941875:

x^2 + 11*x + 29

comment: Cremona labels 50a2 to 50a4.

3625, -263125

25, 1475:

x^2 - 4*x - 21

comment: Cremona labels 50a3 to 50a1.

-26375, 1941875

6025, 467675:

x^2 - 54*x + 329

comment: Cremona labels 50a4 to 50a2.

145, -2105

625, 184375:

x^2 - 1

comment: Cremona labels 50b1 to 50b3.

-1055, 15535

150625, 58459375:

x^2 - 10*x + 9

comment: Cremona labels 50b2 to 50b4.

625, 184375

145, -2105:

x^2 + 5*x - 25

comment: Cremona labels 50b3 to 50b1.

150625, 58459375

-1055, 15535:

x^2 + 55*x + 725

comment: Cremona labels 50b4 to 50b2.

-944, 33128

210736, 96740648:

x^2 - 13*x + 22

comment: Cremona labels 57c1 to 57c2.

210736, 96740648

-944, 33128:

x^2 + 69*x + 5861/5

comment: Cremona labels 57c2 to 57c1.

-239, -6137

21841, 3249703:

x^2 - 2*x - 3

comment: Cremona labels 58b1 to 58b2.

21841, 3249703

-239, -6137:

x^2 + 15*x + 71/5

comment: Cremona labels 58b2 to 58b1.

2161, -73225

483121, 335802455:

x^2 - 24*x + 108

comment: Cremona labels 66c1 to 66c3.

-5519, -476425

482641, 336503015:

x^2 - 14*x - 32

comment: Cremona labels 66c2 to 66c4.

483121, 335802455

2161, -73225:

x^2 + 121*x + 18271/5

comment: Cremona labels 66c3 to 66c1.

482641, 336503015

-5519, -476425:

x^2 + 71*x + 2641/5

comment: Cremona labels 66c4 to 66c2.

400, 8200

-2000, -395000:

x^2 + x - 1

comment: Cremona labels 75a1 to 75a2.

-2000, -395000

400, 8200:

x^2 - 9*x - 36

comment: Cremona labels 75a2 to 75a1.

-80, -3160

10000, 1025000:

x^2 - x - 2

comment: Cremona labels 75c1 to 75c2.

10000, 1025000

-80, -3160:

x^2 + 9*x - 11

comment: Cremona labels 75c2 to 75c1.

144, 4104

4464, -540216:

x^2 + x - 2

comment: Cremona labels 99d1 to 99d2.

4464, -540216

144, 4104:

x^2 - 5*x - 241/5

comment: Cremona labels 99d2 to 99d1.

4464, -540216

3378384, -6209598456:

x^2 + 61*x + 658

comment: Cremona labels 99d2 to 99d3.

3378384, -6209598456

4464, -540216:

x^2 - 305*x + 116279/5

comment: Cremona labels 99d3 to 99d2.

110

-479, 42319

285121, 152245279:

x^2 - 16*x + 39

comment: Cremona labels 110a1 to 110a2.

110

285121, 152245279

-479, 42319:

x^2 + 85*x + 9001/5

comment: Cremona labels 110a2 to 110a1.

118

1201, -42857

-5519, 2184823:

x^2 - 2*x - 3

comment: Cremona labels 118b1 to 118b2.

118

-5519, 2184823

1201, -42857:

x^2 + 15*x - 589/5

comment: Cremona labels 118b2 to 118b1.

121

1441, 54703

14641, -6925193:

x^5 + 14*x^4 + 63*x^3 + 62*x^2 - 230*x - 439

comment: Cremona labels 121a1 to 121a2.

121

14641, -6925193

1441, 54703:

x^5 - 129*x^4 + 800*x^3 + 81847*x^2 - 421871*x - 4132831

comment: Cremona labels 121a2 to 121a1.

121

352, -6776

42592, 9018856:

x^5 - 9*x^4 + 17*x^3 + 20*x^2 - 73*x + 43

comment: Cremona labels 121b1 to 121b2.

121

42592, 9018856

352, -6776:

x^5 + 79*x^4 + 633*x^3 - 42440*x^2 - 781161*x - 3183269

comment: Cremona labels 121b2 to 121b1.

121

121, 5203

174361, -72809693:

x^5 + 14*x^4 + 30*x^3 - 37*x^2 - 76*x + 1

comment: Cremona labels 121c1 to 121c2.

121

174361, -72809693

121, 5203:

x^5 - 129*x^4 + 4793*x^3 + 9973*x^2 - 3694800*x + 52660939

comment: Cremona labels 121c2 to 121c1.

121

1936, 202312

60016, -26630648:

x^2 + 3*x - 28

comment: Cremona labels 121d1 to 121d2.

121

60016, -26630648

1936, 202312:

x^2 - 19*x - 3209/5

comment: Cremona labels 121d2 to 121d1.

121

60016, -26630648

45420496, -306110205368:

x^2 + 223*x + 8772

comment: Cremona labels 121d2 to 121d3.

121

45420496, -306110205368

60016, -26630648:

x^2 - 1119*x + 1565171/5

comment: Cremona labels 121d3 to 121d2.

123

496, -11800

-944, 774440:

x^2 - x - 2

comment: Cremona labels 123a1 to 123a2.

123

-944, 774440

496, -11800:

x^2 + 9*x - 319/5

comment: Cremona labels 123a2 to 123a1.

147

5488, -441784

2145808, 3145828616:

x^6 - 47*x^5 + 263*x^4 + 4761*x^3 - 43363*x^2 + 61051*x + 128143

comment: Cremona labels 147b1 to 147b2.

147

2145808, 3145828616

5488, -441784:

x^6 + 639*x^5 + 105564*x^4 - 4898767*x^3 - 2876210181*x^2 - 262270607280*x - 97458113406809/13

comment: Cremona labels 147b2 to 147b1.

147

112, 1288

43792, -9171512:

x^6 + 5*x^5 - 3*x^4 - 15*x^3 - 5*x^2 + 3*x + 1

comment: Cremona labels 147c1 to 147c2.

147

43792, -9171512

112, 1288:

x^6 - 93*x^5 + 2286*x^4 + 11745*x^3 - 1209087*x^2 + 16292664*x - 887616549/13

comment: Cremona labels 147c2 to 147c1.

150

145, 2375

1345, -237025:

x^2 + x - 1

comment: Cremona labels 150a1 to 150a3.

150

2545, 128375

39745, -7897825:

x^2 + 11*x + 29

comment: Cremona labels 150a2 to 150a4.

150

1345, -237025

145, 2375:

x^2 - 4*x - 32

comment: Cremona labels 150a3 to 150a1.

150

39745, -7897825

2545, 128375:

x^2 - 54*x + 648

comment: Cremona labels 150a4 to 150a2.

150

3625, 296875

33625, -29628125:

x^2 + 5*x - 25

comment: Cremona labels 150b1 to 150b3.

150

63625, 16046875

993625, -987228125:

x^2 + 55*x + 725

comment: Cremona labels 150b2 to 150b4.

150

33625, -29628125

3625, 296875:

x^2 - 20*x - 800

comment: Cremona labels 150b3 to 150b1.

150

993625, -987228125

63625, 16046875:

x^2 - 270*x + 16200

comment: Cremona labels 150b4 to 150b2.

155

-464, -8216

40336, 8116264:

x^2 - 5*x

comment: Cremona labels 155a1 to 155a2.

155

40336, 8116264

-464, -8216:

x^2 + 21*x + 311/5

comment: Cremona labels 155a2 to 155a1.

158

20161, -2837537

1122241, 1188820063:

x^2 - 42*x + 377

comment: Cremona labels 158c1 to 158c2.

158

1122241, 1188820063

20161, -2837537:

x^2 + 215*x + 56221/5

comment: Cremona labels 158c2 to 158c1.

162

225, -3537

4545, 622431:

x^3 - 3*x^2 + 3

comment: Cremona labels 162b1 to 162b3.

162

-1215, -6561

465345, 317440863:

x^3 - 21*x^2 + 66*x - 19

comment: Cremona labels 162b2 to 162b4.

162

4545, 622431

225, -3537:

x^3 + 15*x^2 - 117*x - 4821/7

comment: Cremona labels 162b3 to 162b1.

162

465345, 317440863

-1215, -6561:

x^3 + 141*x^2 + 6303*x + 604613/7

comment: Cremona labels 162b4 to 162b2.

162

-135, 243

51705, -11757069:

x^3 + 6*x^2 + 3*x - 1

comment: Cremona labels 162c1 to 162c3.

162

2025, 95499

40905, -16805637:

x^3 + 6*x^2 - 15*x - 73

comment: Cremona labels 162c2 to 162c4.

162

51705, -11757069

-135, 243:

x^3 - 48*x^2 + 732*x - 24064/7

comment: Cremona labels 162c3 to 162c1.

162

40905, -16805637

2025, 95499:

x^3 - 48*x^2 - 960*x + 137216/7

comment: Cremona labels 162c4 to 162c2.

171

-8496, -894456

1896624, -2611997496:

x^2 + 41*x + 238

comment: Cremona labels 171c1 to 171c2.

171

1896624, -2611997496

-8496, -894456:

x^2 - 205*x + 51719/5

comment: Cremona labels 171c2 to 171c1.

174

49, -118441

312529, 175228199:

x^3 - 12*x^2 - 36*x + 112

comment: Cremona labels 174b1 to 174b2.

174

312529, 175228199

49, -118441:

x^3 + 86*x^2 + 503*x - 419467/7

comment: Cremona labels 174b2 to 174b1.

175

-80, 1000

7120, -603800:

x^2 + x - 1

comment: Cremona labels 175a1 to 175a2.

175

7120, -603800

-80, 1000:

x^2 - 9*x + 8

comment: Cremona labels 175a2 to 175a1.

175

-2000, 125000

178000, -75475000:

x^2 + 9*x - 11

comment: Cremona labels 175c1 to 175c2.

175

178000, -75475000

-2000, 125000:

x^2 - 41*x + 114

comment: Cremona labels 175c2 to 175c1.

176

256, 9728

7936, -1280512:

x^2 + 2*x - 3

comment: Cremona labels 176b1 to 176b2.

176

7936, -1280512

256, 9728:

x^2 - 6*x - 439/5

comment: Cremona labels 176b2 to 176b1.

176

7936, -1280512

6006016, -14719048192:

x^2 + 82*x + 1197

comment: Cremona labels 176b2 to 176b3.

176

6006016, -14719048192

7936, -1280512:

x^2 - 406*x + 206041/5

comment: Cremona labels 176b3 to 176b2.

186

-719, -6697

66961, 17335943:

x^2 - 6*x

comment: Cremona labels 186b1 to 186b2.

186

66961, 17335943

-719, -6697:

x^2 + 31*x + 961/5

comment: Cremona labels 186b2 to 186b1.

198

19449, 1977075

4348089, -9066666285:

x^2 + 72*x + 972

comment: Cremona labels 198e1 to 198e3.

198

-49671, 12863475

4343769, -9085581405:

x^2 + 42*x - 288

comment: Cremona labels 198e2 to 198e4.

198

4348089, -9066666285

19449, 1977075:

x^2 - 363*x + 164439/5

comment: Cremona labels 198e3 to 198e1.

198

4343769, -9085581405

-49671, 12863475:

x^2 - 213*x + 23769/5

comment: Cremona labels 198e4 to 198e2.

203

-944, 1000

103216, 33164200:

x^2 - 9*x + 8

comment: Cremona labels 203a1 to 203a2.

203

103216, 33164200

-944, 1000:

x^2 + 41*x + 1891/5

comment: Cremona labels 203a2 to 203a1.

208

2064, 143424

163344, -72432576:

x^3 + 9*x^2 - 37*x - 165

comment: Cremona labels 208d1 to 208d2.

208

163344, -72432576

2064, 143424:

x^3 - 63*x^2 - 1381*x + 473269/7

comment: Cremona labels 208d2 to 208d1.

225

-720, 85320

90000, -27675000:

x^2 + 5*x - 14

comment: Cremona labels 225d1 to 225d2.

225

90000, -27675000

-720, 85320:

x^2 - 25*x - 125

comment: Cremona labels 225d2 to 225d1.

225

3600, -221400

-18000, 10665000:

x^2 - 5*x - 5

comment: Cremona labels 225e1 to 225e2.

225

-18000, 10665000

3600, -221400:

x^2 + 25*x - 350

comment: Cremona labels 225e2 to 225e1.

234

1161, 60507

91881, -30557493:

x^3 + 6*x^2 - 24*x - 64

comment: Cremona labels 234a1 to 234a2.

234

91881, -30557493

1161, 60507:

x^3 - 48*x^2 - 753*x + 200999/7

comment: Cremona labels 234a2 to 234a1.

246

8401, 24047207

27817681, 146717319527:

x^2 - 172*x + 5092

comment: Cremona labels 246b1 to 246b2.

246

27817681, 146717319527

8401, 24047207:

x^2 + 861*x + 926231/5

comment: Cremona labels 246b2 to 246b1.

258

-7631, -1489321

2875249, 4876010999:

x^3 - 42*x^2 - 36*x + 1512

comment: Cremona labels 258f1 to 258f2.

258

2875249, 4876010999

-7631, -1489321:

x^3 + 296*x^2 + 21193*x - 627017/7

comment: Cremona labels 258f2 to 258f1.

275

400, -19000

12400, 2501000:

x^2 - x - 6

comment: Cremona labels 275b1 to 275b2.

275

12400, 2501000

400, -19000:

x^2 + 9*x - 131

comment: Cremona labels 275b2 to 275b1.

275

12400, 2501000

9384400, 28748141000:

x^2 - 101*x + 1794

comment: Cremona labels 275b2 to 275b3.

275

9384400, 28748141000

12400, 2501000:

x^2 + 509*x + 64769

comment: Cremona labels 275b3 to 275b2.

286

-13439, -238913

1340641, 1552650607:

x^2 - 28*x + 27

comment: Cremona labels 286d1 to 286d2.

286

1340641, 1552650607

-13439, -238913:

x^2 + 145*x + 22621/5

comment: Cremona labels 286d2 to 286d1.

294

2401, -271313

331681, 198185743:

x^3 - 11*x^2 - 74*x + 281

comment: Cremona labels 294a1 to 294a2.

294

331681, 198185743

2401, -271313:

x^3 + 87*x^2 - 1593*x - 149855

comment: Cremona labels 294a2 to 294a1.

294

49, 791

6769, -577801:

x^3 + 2*x^2 - x - 1

comment: Cremona labels 294b1 to 294b2.

294

6769, -577801

49, 791:

x^3 - 12*x^2 - 36*x + 432

comment: Cremona labels 294b2 to 294b1.

Definition

Let $E/\mathbb Q$ be recovered from $(c_4,c_6)$ by Formula (1). For a rational isogeny $\varphi:E\to E'$ of prime degree $\ell\ge5$, the entry is the monic polynomial $K_\varphi(x)$ whose roots are the $x$-coordinates of the nonzero points of $\ker\varphi$ [1].

Parameters

$N$

— conductor of the source curve $E$ ($N\geq1$)

$c_4$

— invariant $c_4$ of the reduced global minimal model of $E$

$c_6$

— invariant $c_6$ of the reduced global minimal model of $E$

$\ell$

— isogeny degree ($\ell$ is prime and $\ell\geq5$)

$c'_4$

— invariant $c_4$ of the target curve $E'$

$c'_6$

— invariant $c_6$ of the target curve $E'$

Formulas

(1)

For the reduced models in this table, $b_2$ is the unique element of $\{-4,-3,0,1,4,5\}$ with $b_2\equiv -c_6\pmod{12}$, $b_4=(b_2^2-c_4)/24$, and $b_6=(-c_6-b_2^3+36b_2b_4)/216$. The model has $a_1\equiv b_2\pmod 2$, $a_3\equiv b_6\pmod 2$ with $a_1,a_3\in\{0,1\}$ and $a_2\in\{-1,0,1\}$, and $a_2=(b_2-a_1^2)/4$, $a_4=(b_4-a_1a_3)/2$ and $a_6=(b_6-a_3^2)/4$.

(2)

For prime $\ell\geq5$, $K_\varphi(x)$ has degree $(\ell-1)/2$ and divides the division polynomial $\psi_\ell(x)$ in $\mathbb{Q}[x]$. Since $\psi_\ell(x)$ has leading coefficient $\ell$, a monic kernel polynomial need not have integer coefficients.

Comments

(3)

The source curve is identified by $N$, $c_4$ and $c_6$. The target invariants $c'_4$ and $c'_6$ distinguish rational isogenies of the same degree from the same source curve. Each entry comment gives the source and target Cremona labels. LMFDB [2] numbers curves within an isogeny class differently from Cremona's tables [3], so those labels are not LMFDB labels.

(4)

Degree $2$ and degree $3$ rational isogenies are not listed. Their kernel polynomials are linear, so each is a single rational $x$-coordinate written as a polynomial.

(5)

A kernel polynomial can be used as input to Velu's formulas for constructing $\varphi$ and $E'$ [6].

Programs

(P1)

Sage

E = EllipticCurve('11a1')
for phi in E.isogenies_prime_degree(5):
    print(phi.codomain().cremona_label(), phi.kernel_polynomial())

Links

[1]

LMFDB knowl: Isogenies of elliptic curves

[2]

LMFDB: Elliptic curves over Q

[3]

SageMath reference: Cremona's tables of elliptic curves

[4]

SageMath reference: isogenies of elliptic curves

[5]

PARI/GP manual: ellisogeny

[6]

SageMath reference: elliptic-curve isogenies and Velu's formulas

Similar tables

Division polynomials $\psi_n$ of elliptic curves over $\mathbb{Q}$ — stores the division polynomial that each listed kernel polynomial divides

Modular polynomials for the $j$-invariant — stores modular equations whose roots determine which curves are $\ell$-isogenous

Canonical modular polynomials $\Phi^c_\ell$ — stores smaller modular equations for the same prime-degree isogeny relation

Weil polynomials of abelian varieties over prime fields — indexes isogeny classes of abelian varieties over finite fields rather than explicit kernels of isogenies over $\mathbb{Q}$

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds every rational isogeny of prime degree $\ell\geq5$ whose source curve has conductor $N\leq300$, the small-conductor range selected for this table; Sage's mini Cremona database [3] is complete in this range)

Sources of data: [3], [4]

How they were obtained:

The generator reads exact Weierstrass coefficients from Sage's mini Cremona database [3], enumerates Sage's exact rational isogenies of prime degree $\ell\geq5$ [4], and stores each monic kernel polynomial in $\mathbb{Q}[x]$.

It verifies every row has degree $(\ell-1)/2$ and divides the exact division polynomial $\psi_\ell$ of the source curve. It also checks, using the source curve and the stored kernel polynomial rather than Sage's isogeny object, that PARI/GP's ellisogeny [5] returns a curve whose $(c_4,c_6)$ invariants are the target parameters. On the rows where the division-polynomial table contains the same source curve and $\ell=5$, it verifies divisibility against that stored row too.