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display: $D$ constraints: positive fundamental discriminant- conductor:+ label: type: Symbolic- title: LMFDB conductor label- display: conductor- class:- type: Symbolic- title: isogeny class label- display: class- curve:- type: Z- title: curve number in the LMFDB class- display: curve+ title: LMFDB curve label, without the field prefix+ display: label+ constraints: 'the LMFDB curve label with the field label $2.2.D.1$ removed: `31.1-a1`+ for the curve `2.2.5.1-31.1-a1`' Comments:- comment-index: The row with $D=12$, conductor label $47.1$, isogeny class $b$, and- curve number $1$ is the LMFDB CITE{LMFDB} curve 2.2.12.1-47.1-b1. There is one- row per curve rather than one row per isogeny class, since the regulator is attached- to the Mordell-Weil lattice of the curve. A curve and its Galois conjugate have- the same regulator, so the rows for conductors $51.2$ and $51.3$ over $D=13$ repeat- one another with the curve numbers exchanged. Curves in one isogeny class have- regulators in a rational ratio; in class $17.1$-a over $D=17$ the ratios are powers- of $2$.+ comment-index: 'The label of a row is the LMFDB CITE{LMFDB} curve label with the+ field label removed: the row with $D=12$ and label `47.1-b1` is the curve `2.2.12.1-47.1-b1`,+ where `47.1` labels the conductor ideal, of norm $47$, `b` is the isogeny class+ and `1` the curve in it. There is one row per curve rather than one row per isogeny+ class, since the regulator is attached to the Mordell-Weil lattice of the curve.+ A curve and its Galois conjugate have the same regulator, so over $D=13$ the rows+ whose conductor ideals are labelled `51.2` and `51.3` repeat one another with+ the curve numbers exchanged. Curves in one isogeny class have regulators in a+ rational ratio; in the class `17.1-a` over $D=17$ the ratios are powers of $2$.' comment-field-generator: 'The generator $w$ of $K$ follows the LMFDB CITE{LMFDB} convention used in ecnf-data CITE{ecnf-data}: it satisfies $w^2-w-1=0$ for $D=5$,
group parameters: - - D- - - conductor- - - class- - - curve+ - - label Numbers:-- params:- D: '12'- conductor: '47.1'- class: b- curve: '1'- number: '0.098568841514602893460165344842850895'- comment: LMFDB curve 2.2.12.1-47.1-b1 has conductor ideal $(-4w-1)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}+w{x}^{2}+\left(w-1\right){x}$.-- params:- D: '12'- conductor: '47.1'- class: b- curve: '2'- number: '0.049284420757301446730082672421425447'- comment: LMFDB curve 2.2.12.1-47.1-b2 has conductor ideal $(-4w-1)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}+w{x}^{2}+\left(6w-11\right){x}-14w+24$.-- params:- D: '12'- conductor: '47.2'- class: b- curve: '1'- number: '0.098568841514602893460165344842850895'- comment: LMFDB curve 2.2.12.1-47.2-b1 has conductor ideal $(4w-1)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}-w{x}^{2}+\left(-w-1\right){x}$.-- params:- D: '12'- conductor: '47.2'- class: b- curve: '2'- number: '0.049284420757301446730082672421425447'- comment: LMFDB curve 2.2.12.1-47.2-b2 has conductor ideal $(4w-1)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}-w{x}^{2}+\left(-6w-11\right){x}+14w+24$.-- params:- D: '13'- conductor: '51.2'- class: a- curve: '1'- number: '0.034450165062797493363476807161562966'- comment: LMFDB curve 2.2.13.1-51.2-a1 has conductor ideal $(-2w+9)$, rank $1$, and- equation ${y}^2+w{y}={x}^{3}-{x}^{2}+\left(-2w+4\right){x}+w-3$.-- params:- D: '13'- conductor: '51.2'- class: a- curve: '2'- number: '0.011483388354265831121158935720520988'- comment: LMFDB curve 2.2.13.1-51.2-a2 has conductor ideal $(-2w+9)$, rank $1$, and- equation ${y}^2+w{y}={x}^{3}-{x}^{2}+\left(w-1\right){x}-w+1$.-- params:- D: '13'- conductor: '51.3'- class: a- curve: '1'- number: '0.011483388354265831121158935720520988'- comment: LMFDB curve 2.2.13.1-51.3-a1 has conductor ideal $(2w+7)$, rank $1$, and- equation ${y}^2+\left(w+1\right){y}={x}^{3}-{x}^{2}-w{x}$.-- params:- D: '13'- conductor: '51.3'- class: a- curve: '2'- number: '0.034450165062797493363476807161562966'- comment: LMFDB curve 2.2.13.1-51.3-a2 has conductor ideal $(2w+7)$, rank $1$, and- equation ${y}^2+\left(w+1\right){y}={x}^{3}-{x}^{2}+\left(2w+2\right){x}-2w-2$.-- params:- D: '13'- conductor: '52.1'- class: b- curve: '1'- number: '1.3834480278188076609662078599889399'- comment: LMFDB curve 2.2.13.1-52.1-b1 has conductor ideal $(-4w+2)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-213{x}-1257$.-- params:- D: '13'- conductor: '52.1'- class: b- curve: '2'- number: '0.19763543254554395156660112285556285'- comment: LMFDB curve 2.2.13.1-52.1-b2 has conductor ideal $(-4w+2)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-3{x}+3$.-- params:- D: '13'- conductor: '53.1'- class: a- curve: '1'- number: '0.047438850434816421561347347750918711'- comment: LMFDB curve 2.2.13.1-53.1-a1 has conductor ideal $(w+7)$, rank $1$, and- equation ${y}^2+w{x}{y}+{y}={x}^{3}-w{x}^{2}+\left(-w+1\right){x}$.-- params:- D: '13'- conductor: '53.2'- class: a- curve: '1'- number: '0.047438850434816421561347347750918711'- comment: LMFDB curve 2.2.13.1-53.2-a1 has conductor ideal $(w-8)$, rank $1$, and- equation ${y}^2+\left(w+1\right){x}{y}+{y}={x}^{3}-{x}^{2}$.-- params:- D: '17'- conductor: '17.1'- class: a- curve: '1'- number: '6.0705123554606643216641072065434946'- comment: LMFDB curve 2.2.17.1-17.1-a1 has conductor ideal $(-2w+1)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(330w-936\right){x}+4996w-13110$.-- params:- D: '17'- conductor: '17.1'- class: a- curve: '2'- number: '0.75881404443258304020801340081793683'- comment: LMFDB curve 2.2.17.1-17.1-a2 has conductor ideal $(-2w+1)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}-14$.-- params:- D: '17'- conductor: '17.1'- class: a- curve: '3'- number: '0.37940702221629152010400670040896841'- comment: LMFDB curve 2.2.17.1-17.1-a3 has conductor ideal $(-2w+1)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(6w+9\right){x}+4w+6$.-- params:- D: '17'- conductor: '17.1'- class: a- curve: '4'- number: '0.75881404443258304020801340081793683'- comment: LMFDB curve 2.2.17.1-17.1-a4 has conductor ideal $(-2w+1)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}$.-- params:- D: '17'- conductor: '17.1'- class: a- curve: '5'- number: '1.5176280888651660804160268016358736'- comment: LMFDB curve 2.2.17.1-17.1-a5 has conductor ideal $(-2w+1)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-6{x}-4$.-- params:- D: '17'- conductor: '17.1'- class: a- curve: '6'- number: '0.37940702221629152010400670040896841'- comment: LMFDB curve 2.2.17.1-17.1-a6 has conductor ideal $(-2w+1)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-6w+15\right){x}-4w+10$.-- params:- D: '17'- conductor: '17.1'- class: a- curve: '7'- number: '3.0352561777303321608320536032717473'- comment: LMFDB curve 2.2.17.1-17.1-a7 has conductor ideal $(-2w+1)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-91{x}-310$.-- params:- D: '17'- conductor: '17.1'- class: a- curve: '8'- number: '6.0705123554606643216641072065434946'- comment: LMFDB curve 2.2.17.1-17.1-a8 has conductor ideal $(-2w+1)$, rank $1$, and- equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-330w-606\right){x}-4996w-8114$.-- params:- D: '17'- conductor: '38.2'- class: a- curve: '1'- number: '0.085387436507748877571154453414499536'- comment: LMFDB curve 2.2.17.1-38.2-a1 has conductor ideal $(-3w+2)$, rank $1$, and- equation ${y}^2+{x}{y}={x}^{3}+\left(-w+1\right){x}^{2}+\left(3w-7\right){x}-2w+5$.-- params:- D: '17'- conductor: '38.2'- class: a- curve: '2'- number: '0.17077487301549775514230890682899907'- comment: LMFDB curve 2.2.17.1-38.2-a2 has conductor ideal $(-3w+2)$, rank $1$, and- equation ${y}^2+{x}{y}+w{y}={x}^{3}+{x}^{2}+\left(-4w-5\right){x}+w+1$.-- params:- D: '17'- conductor: '38.2'- class: a- curve: '3'- number: '0.34154974603099551028461781365799814'- comment: LMFDB curve 2.2.17.1-38.2-a3 has conductor ideal $(-3w+2)$, rank $1$, and- equation ${y}^2+{x}{y}+\left(w+1\right){y}={x}^{3}+\left(w-5\right){x}+w-6$.-- params:- D: '17'- conductor: '38.2'- class: a- curve: '4'- number: '0.085387436507748877571154453414499536'- comment: LMFDB curve 2.2.17.1-38.2-a4 has conductor ideal $(-3w+2)$, rank $1$, and- equation ${y}^2+{x}{y}+w{y}={x}^{3}+{x}^{2}+\left(-49w-75\right){x}+243w+379$.-- params:- D: '17'- conductor: '38.3'- class: a- curve: '1'- number: '0.085387436507748877571154453414499536'- comment: LMFDB curve 2.2.17.1-38.3-a1 has conductor ideal $(3w-1)$, rank $1$, and- equation ${y}^2+{x}{y}+\left(w+1\right){y}={x}^{3}+{x}^{2}+\left(48w-124\right){x}-244w+622$.-- params:- D: '17'- conductor: '38.3'- class: a- curve: '2'- number: '0.17077487301549775514230890682899907'- comment: LMFDB curve 2.2.17.1-38.3-a2 has conductor ideal $(3w-1)$, rank $1$, and- equation ${y}^2+{x}{y}+\left(w+1\right){y}={x}^{3}+{x}^{2}+\left(3w-9\right){x}-2w+2$.-- params:- D: '17'- conductor: '38.3'- class: a- curve: '3'- number: '0.085387436507748877571154453414499536'- comment: LMFDB curve 2.2.17.1-38.3-a3 has conductor ideal $(3w-1)$, rank $1$, and- equation ${y}^2+{x}{y}={x}^{3}+w{x}^{2}+\left(-3w-4\right){x}+2w+3$.-- params:- D: '17'- conductor: '38.3'- class: a- curve: '4'- number: '0.34154974603099551028461781365799814'- comment: LMFDB curve 2.2.17.1-38.3-a4 has conductor ideal $(3w-1)$, rank $1$, and- equation ${y}^2+{x}{y}+w{y}={x}^{3}+\left(-2w-3\right){x}-2w-4$.-- params:- D: '17'- conductor: '47.1'- class: a- curve: '1'- number: '0.047398169405353604132198011561955181'- comment: LMFDB curve 2.2.17.1-47.1-a1 has conductor ideal $(-2w+9)$, rank $1$, and- equation ${y}^2+{y}={x}^{3}+\left(w-1\right){x}^{2}+{x}$.-- params:- D: '17'- conductor: '47.2'- class: a- curve: '1'- number: '0.047398169405353604132198011561955181'- comment: LMFDB curve 2.2.17.1-47.2-a1 has conductor ideal $(2w+7)$, rank $1$, and- equation ${y}^2+{y}={x}^{3}-w{x}^{2}+{x}$.-- params:- D: '17'- conductor: '52.3'- class: b- curve: '1'- number: '0.011548536804434145224328626608586774'- comment: LMFDB curve 2.2.17.1-52.3-b1 has conductor ideal $(w+7)$, rank $1$, and- equation ${y}^2+\left(w+1\right){y}={x}^{3}+\left(-16w-25\right){x}+50w+78$.-- params:- D: '17'- conductor: '52.6'- class: b- curve: '1'- number: '0.011548536804434145224328626608586774'- comment: LMFDB curve 2.2.17.1-52.6-b1 has conductor ideal $(w-8)$, rank $1$, and- equation ${y}^2+w{y}={x}^{3}+\left(16w-41\right){x}-51w+129$.+ '12':+ 47.1-b1:+ number: '0.098568841514602893460165344842850895'+ comment: LMFDB curve 2.2.12.1-47.1-b1 has conductor ideal $(-4w-1)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}+w{x}^{2}+\left(w-1\right){x}$.+ 47.1-b2:+ number: '0.049284420757301446730082672421425447'+ comment: LMFDB curve 2.2.12.1-47.1-b2 has conductor ideal $(-4w-1)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}+w{x}^{2}+\left(6w-11\right){x}-14w+24$.+ 47.2-b1:+ number: '0.098568841514602893460165344842850895'+ comment: LMFDB curve 2.2.12.1-47.2-b1 has conductor ideal $(4w-1)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}-w{x}^{2}+\left(-w-1\right){x}$.+ 47.2-b2:+ number: '0.049284420757301446730082672421425447'+ comment: LMFDB curve 2.2.12.1-47.2-b2 has conductor ideal $(4w-1)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}-w{x}^{2}+\left(-6w-11\right){x}+14w+24$.+ '13':+ 51.2-a1:+ number: '0.034450165062797493363476807161562966'+ comment: LMFDB curve 2.2.13.1-51.2-a1 has conductor ideal $(-2w+9)$, rank $1$,+ and equation ${y}^2+w{y}={x}^{3}-{x}^{2}+\left(-2w+4\right){x}+w-3$.+ 51.2-a2:+ number: '0.011483388354265831121158935720520988'+ comment: LMFDB curve 2.2.13.1-51.2-a2 has conductor ideal $(-2w+9)$, rank $1$,+ and equation ${y}^2+w{y}={x}^{3}-{x}^{2}+\left(w-1\right){x}-w+1$.+ 51.3-a1:+ number: '0.011483388354265831121158935720520988'+ comment: LMFDB curve 2.2.13.1-51.3-a1 has conductor ideal $(2w+7)$, rank $1$,+ and equation ${y}^2+\left(w+1\right){y}={x}^{3}-{x}^{2}-w{x}$.+ 51.3-a2:+ number: '0.034450165062797493363476807161562966'+ comment: LMFDB curve 2.2.13.1-51.3-a2 has conductor ideal $(2w+7)$, rank $1$,+ and equation ${y}^2+\left(w+1\right){y}={x}^{3}-{x}^{2}+\left(2w+2\right){x}-2w-2$.+ 52.1-b1:+ number: '1.3834480278188076609662078599889399'+ comment: LMFDB curve 2.2.13.1-52.1-b1 has conductor ideal $(-4w+2)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-213{x}-1257$.+ 52.1-b2:+ number: '0.19763543254554395156660112285556285'+ comment: LMFDB curve 2.2.13.1-52.1-b2 has conductor ideal $(-4w+2)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-3{x}+3$.+ 53.1-a1:+ number: '0.047438850434816421561347347750918711'+ comment: LMFDB curve 2.2.13.1-53.1-a1 has conductor ideal $(w+7)$, rank $1$,+ and equation ${y}^2+w{x}{y}+{y}={x}^{3}-w{x}^{2}+\left(-w+1\right){x}$.+ 53.2-a1:+ number: '0.047438850434816421561347347750918711'+ comment: LMFDB curve 2.2.13.1-53.2-a1 has conductor ideal $(w-8)$, rank $1$,+ and equation ${y}^2+\left(w+1\right){x}{y}+{y}={x}^{3}-{x}^{2}$.+ '17':+ 17.1-a1:+ number: '6.0705123554606643216641072065434946'+ comment: LMFDB curve 2.2.17.1-17.1-a1 has conductor ideal $(-2w+1)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(330w-936\right){x}+4996w-13110$.+ 17.1-a2:+ number: '0.75881404443258304020801340081793683'+ comment: LMFDB curve 2.2.17.1-17.1-a2 has conductor ideal $(-2w+1)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}-14$.+ 17.1-a3:+ number: '0.37940702221629152010400670040896841'+ comment: LMFDB curve 2.2.17.1-17.1-a3 has conductor ideal $(-2w+1)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(6w+9\right){x}+4w+6$.+ 17.1-a4:+ number: '0.75881404443258304020801340081793683'+ comment: LMFDB curve 2.2.17.1-17.1-a4 has conductor ideal $(-2w+1)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}$.+ 17.1-a5:+ number: '1.5176280888651660804160268016358736'+ comment: LMFDB curve 2.2.17.1-17.1-a5 has conductor ideal $(-2w+1)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-6{x}-4$.+ 17.1-a6:+ number: '0.37940702221629152010400670040896841'+ comment: LMFDB curve 2.2.17.1-17.1-a6 has conductor ideal $(-2w+1)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-6w+15\right){x}-4w+10$.+ 17.1-a7:+ number: '3.0352561777303321608320536032717473'+ comment: LMFDB curve 2.2.17.1-17.1-a7 has conductor ideal $(-2w+1)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-91{x}-310$.+ 17.1-a8:+ number: '6.0705123554606643216641072065434946'+ comment: LMFDB curve 2.2.17.1-17.1-a8 has conductor ideal $(-2w+1)$, rank $1$,+ and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-330w-606\right){x}-4996w-8114$.+ 38.2-a1:+ number: '0.085387436507748877571154453414499536'+ comment: LMFDB curve 2.2.17.1-38.2-a1 has conductor ideal $(-3w+2)$, rank $1$,+ and equation ${y}^2+{x}{y}={x}^{3}+\left(-w+1\right){x}^{2}+\left(3w-7\right){x}-2w+5$.+ 38.2-a2:+ number: '0.17077487301549775514230890682899907'+ comment: LMFDB curve 2.2.17.1-38.2-a2 has conductor ideal $(-3w+2)$, rank $1$,+ and equation ${y}^2+{x}{y}+w{y}={x}^{3}+{x}^{2}+\left(-4w-5\right){x}+w+1$.+ 38.2-a3:+ number: '0.34154974603099551028461781365799814'+ comment: LMFDB curve 2.2.17.1-38.2-a3 has conductor ideal $(-3w+2)$, rank $1$,+ and equation ${y}^2+{x}{y}+\left(w+1\right){y}={x}^{3}+\left(w-5\right){x}+w-6$.+ 38.2-a4:+ number: '0.085387436507748877571154453414499536'+ comment: LMFDB curve 2.2.17.1-38.2-a4 has conductor ideal $(-3w+2)$, rank $1$,+ and equation ${y}^2+{x}{y}+w{y}={x}^{3}+{x}^{2}+\left(-49w-75\right){x}+243w+379$.+ 38.3-a1:+ number: '0.085387436507748877571154453414499536'+ comment: LMFDB curve 2.2.17.1-38.3-a1 has conductor ideal $(3w-1)$, rank $1$,+ and equation ${y}^2+{x}{y}+\left(w+1\right){y}={x}^{3}+{x}^{2}+\left(48w-124\right){x}-244w+622$.+ 38.3-a2:+ number: '0.17077487301549775514230890682899907'+ comment: LMFDB curve 2.2.17.1-38.3-a2 has conductor ideal $(3w-1)$, rank $1$,+ and equation ${y}^2+{x}{y}+\left(w+1\right){y}={x}^{3}+{x}^{2}+\left(3w-9\right){x}-2w+2$.+ 38.3-a3:+ number: '0.085387436507748877571154453414499536'+ comment: LMFDB curve 2.2.17.1-38.3-a3 has conductor ideal $(3w-1)$, rank $1$,+ and equation ${y}^2+{x}{y}={x}^{3}+w{x}^{2}+\left(-3w-4\right){x}+2w+3$.+ 38.3-a4:+ number: '0.34154974603099551028461781365799814'+ comment: LMFDB curve 2.2.17.1-38.3-a4 has conductor ideal $(3w-1)$, rank $1$,+ and equation ${y}^2+{x}{y}+w{y}={x}^{3}+\left(-2w-3\right){x}-2w-4$.+ 47.1-a1:+ number: '0.047398169405353604132198011561955181'+ comment: LMFDB curve 2.2.17.1-47.1-a1 has conductor ideal $(-2w+9)$, rank $1$,+ and equation ${y}^2+{y}={x}^{3}+\left(w-1\right){x}^{2}+{x}$.+ 47.2-a1:+ number: '0.047398169405353604132198011561955181'+ comment: LMFDB curve 2.2.17.1-47.2-a1 has conductor ideal $(2w+7)$, rank $1$,+ and equation ${y}^2+{y}={x}^{3}-w{x}^{2}+{x}$.+ 52.3-b1:+ number: '0.011548536804434145224328626608586774'+ comment: LMFDB curve 2.2.17.1-52.3-b1 has conductor ideal $(w+7)$, rank $1$,+ and equation ${y}^2+\left(w+1\right){y}={x}^{3}+\left(-16w-25\right){x}+50w+78$.+ 52.6-b1:+ number: '0.011548536804434145224328626608586774'+ comment: LMFDB curve 2.2.17.1-52.6-b1 has conductor ideal $(w-8)$, rank $1$,+ and equation ${y}^2+w{y}={x}^{3}+\left(16w-41\right){x}-51w+129$.
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