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 |
4.1-a2 |
4.1-a |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 7$ |
3B, 7B.6.3 |
$1$ |
\( 1 \) |
$1$ |
$7.662151750$ |
1.014876791 |
\( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -71396 a - 233865\) , \( 20071185 a + 65731670\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-71396a-233865\right){x}+20071185a+65731670$ |
4.1-d2 |
4.1-d |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 7$ |
3B, 7B.1.3 |
$49$ |
\( 3 \) |
$1$ |
$0.077866687$ |
1.516113114 |
\( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 642628 a - 2747187\) , \( 545795628 a - 2333231135\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(642628a-2747187\right){x}+545795628a-2333231135$ |
36.1-b2 |
36.1-b |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{6} \) |
$1.65254$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B.1.1, 7B.6.3 |
$49$ |
\( 3 \) |
$1$ |
$0.772415932$ |
1.671046829 |
\( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 5674 a - 29613\) , \( -560830 a + 2247997\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(5674a-29613\right){x}-560830a+2247997$ |
36.1-f2 |
36.1-f |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{6} \) |
$1.65254$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B.1.2, 7B.6.3 |
$49$ |
\( 1 \) |
$1$ |
$0.257471977$ |
1.671046829 |
\( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -64690451 a - 211855871\) , \( 550295059293 a + 1802170764457\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-64690451a-211855871\right){x}+550295059293a+1802170764457$ |
128.5-c2 |
128.5-c |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{22} \) |
$2.26923$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B, 7B.6.3 |
$49$ |
\( 2 \cdot 3 \) |
$1$ |
$0.157668741$ |
6.139818101 |
\( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -2811454 a - 9207370\) , \( 4982304576 a + 16316634679\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-2811454a-9207370\right){x}+4982304576a+16316634679$ |
128.5-j2 |
128.5-j |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{22} \) |
$2.26923$ |
$(a-4), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B, 7B.6.3 |
$1$ |
\( 2^{2} \) |
$3.301024792$ |
$0.473006225$ |
1.654505450 |
\( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 1044512 a - 4465409\) , \( -1129233268 a + 4827376713\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(1044512a-4465409\right){x}-1129233268a+4827376713$ |
128.6-c2 |
128.6-c |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{22} \) |
$2.26923$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B, 7B.6.3 |
$49$ |
\( 2 \) |
$1$ |
$0.473006225$ |
6.139818101 |
\( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -115777 a - 381257\) , \( -42294261 a - 138472621\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-115777a-381257\right){x}-42294261a-138472621$ |
128.6-j2 |
128.6-j |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{22} \) |
$2.26923$ |
$(a-4), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B, 7B.6.3 |
$1$ |
\( 2^{2} \) |
$9.903074376$ |
$0.157668741$ |
1.654505450 |
\( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 25303280 a - 108169436\) , \( 134641581530 a - 575581615086\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(25303280a-108169436\right){x}+134641581530a-575581615086$ |
256.1-a2 |
256.1-a |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{28} \) |
$2.69858$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B, 7B.6.3 |
$1$ |
\( 2^{4} \) |
$1$ |
$1.915537937$ |
4.059507167 |
\( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 10282065 a - 43955062\) , \( -34876683146 a + 149094933468\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(10282065a-43955062\right){x}-34876683146a+149094933468$ |
256.1-r2 |
256.1-r |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{28} \) |
$2.69858$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B, 7B.6.3 |
$49$ |
\( 2^{2} \) |
$1$ |
$0.019466671$ |
0.505371038 |
\( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -1142335 a - 3741902\) , \( -1289440094 a - 4222819768\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1142335a-3741902\right){x}-1289440094a-4222819768$ |
*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.