| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 71424.2-a1 |
71424.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{16} \cdot 3^{3} \cdot 31^{2} \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.296057380$ |
$2.356764957$ |
3.222712205 |
\( \frac{355344}{961} a + \frac{314016}{961} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -3 a - 3\) , \( -5 a + 8\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3a-3\right){x}-5a+8$ |
| 71424.2-a2 |
71424.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{8} \cdot 3^{3} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.592114760$ |
$4.713529915$ |
3.222712205 |
\( -\frac{63744}{31} a + \frac{157440}{31} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 2 a + 2\) , \( -3 a + 1\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(2a+2\right){x}-3a+1$ |
| 71424.2-b1 |
71424.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{16} \cdot 3^{9} \cdot 31^{2} \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.540606015$ |
$1.360678882$ |
3.397550170 |
\( \frac{355344}{961} a + \frac{314016}{961} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -21 a + 12\) , \( -24 a - 26\bigr] \) |
${y}^2={x}^{3}+\left(-21a+12\right){x}-24a-26$ |
| 71424.2-b2 |
71424.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{8} \cdot 3^{9} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.081212031$ |
$2.721357765$ |
3.397550170 |
\( -\frac{63744}{31} a + \frac{157440}{31} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 9 a - 3\) , \( -3 a - 5\bigr] \) |
${y}^2={x}^{3}+\left(9a-3\right){x}-3a-5$ |
| 71424.2-c1 |
71424.2-c |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{74} \cdot 3^{6} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \) |
$7.009610537$ |
$0.053178547$ |
3.443417772 |
\( \frac{936087656892551}{1040187392} a - \frac{833285178768245}{1040187392} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -36023 a - 26375\) , \( 4437145 a + 716652\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-36023a-26375\right){x}+4437145a+716652$ |
| 71424.2-c2 |
71424.2-c |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{26} \cdot 3^{6} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2^{2} \) |
$0.280384421$ |
$1.329463692$ |
3.443417772 |
\( \frac{24551}{62} a + \frac{66955}{62} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -23 a + 25\) , \( 25 a + 12\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-23a+25\right){x}+25a+12$ |
| 71424.2-c3 |
71424.2-c |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{34} \cdot 3^{6} \cdot 31^{5} \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5Cs.4.1 |
$1$ |
\( 2^{2} \) |
$1.401922107$ |
$0.265892738$ |
3.443417772 |
\( -\frac{511363962461}{916132832} a + \frac{764718499383}{458066416} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 457 a + 265\) , \( -4055 a + 2508\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(457a+265\right){x}-4055a+2508$ |
| 71424.2-d1 |
71424.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{26} \cdot 3^{6} \cdot 31^{3} \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \) |
$1$ |
$0.727178501$ |
1.679346813 |
\( \frac{10618695}{29791} a - \frac{124425801}{59582} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -75 a - 21\) , \( -480 a + 122\bigr] \) |
${y}^2={x}^{3}+\left(-75a-21\right){x}-480a+122$ |
| 71424.2-d2 |
71424.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{30} \cdot 3^{6} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \) |
$1$ |
$0.727178501$ |
1.679346813 |
\( -\frac{44272737}{124} a + \frac{99194139}{248} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -195 a + 291\) , \( 864 a + 962\bigr] \) |
${y}^2={x}^{3}+\left(-195a+291\right){x}+864a+962$ |
| 71424.2-e1 |
71424.2-e |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{22} \cdot 3^{6} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.274027341$ |
$1.622671209$ |
4.107558749 |
\( \frac{20086}{31} a - \frac{69202}{31} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -2 a + 19\) , \( -49 a + 9\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a+19\right){x}-49a+9$ |
| 71424.2-f1 |
71424.2-f |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{20} \cdot 3^{10} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.724272481$ |
1.672635649 |
\( \frac{1160560100}{279} a - \frac{258451496}{93} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -66 a + 387\) , \( -3009 a + 681\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-66a+387\right){x}-3009a+681$ |
| 71424.2-f2 |
71424.2-f |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{16} \cdot 3^{8} \cdot 31^{2} \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.448544963$ |
1.672635649 |
\( \frac{4637360}{2883} a + \frac{3524368}{2883} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -6 a + 27\) , \( -45 a + 9\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a+27\right){x}-45a+9$ |
| 71424.2-f3 |
71424.2-f |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{20} \cdot 3^{7} \cdot 31^{4} \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.724272481$ |
1.672635649 |
\( -\frac{2499774004}{2770563} a + \frac{4015907912}{2770563} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 54 a - 93\) , \( -297 a + 153\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(54a-93\right){x}-297a+153$ |
| 71424.2-f4 |
71424.2-f |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{8} \cdot 3^{7} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2.897089926$ |
1.672635649 |
\( -\frac{2998016}{93} a + \frac{2572288}{93} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -6 a + 12\) , \( 9 a\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a+12\right){x}+9a$ |
| 71424.2-g1 |
71424.2-g |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{74} \cdot 3^{6} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$25$ |
\( 2 \) |
$1$ |
$0.053178547$ |
3.070264883 |
\( \frac{936087656892551}{1040187392} a - \frac{833285178768245}{1040187392} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 62400 a - 36024\) , \( -4499544 a - 680628\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(62400a-36024\right){x}-4499544a-680628$ |
| 71424.2-g2 |
71424.2-g |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{26} \cdot 3^{6} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$1.329463692$ |
3.070264883 |
\( \frac{24551}{62} a + \frac{66955}{62} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -24\) , \( -24 a + 12\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}-24{x}-24a+12$ |
| 71424.2-g3 |
71424.2-g |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{34} \cdot 3^{6} \cdot 31^{5} \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5Cs.4.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$0.265892738$ |
3.070264883 |
\( -\frac{511363962461}{916132832} a + \frac{764718499383}{458066416} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -720 a + 456\) , \( 4776 a - 2964\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-720a+456\right){x}+4776a-2964$ |
| 71424.2-h1 |
71424.2-h |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{16} \cdot 3^{3} \cdot 31^{2} \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.356764957$ |
2.721357765 |
\( \frac{355344}{961} a + \frac{314016}{961} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -3 a - 3\) , \( 5 a - 8\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-3a-3\right){x}+5a-8$ |
| 71424.2-h2 |
71424.2-h |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.2 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{8} \cdot 3^{3} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (6a-5), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$4.713529915$ |
2.721357765 |
\( -\frac{63744}{31} a + \frac{157440}{31} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 2 a + 2\) , \( 3 a - 1\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+2\right){x}+3a-1$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
