| 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-a1 |
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 |
\( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 71395 a - 305260\) , \( -20071186 a + 85802856\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(71395a-305260\right){x}-20071186a+85802856$ |
| 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-a3 |
4.1-a |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{28} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 7$ |
3B, 7B.6.1 |
$1$ |
\( 1 \) |
$1$ |
$7.662151750$ |
1.014876791 |
\( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -215 a + 920\) , \( 2686 a - 11488\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-215a+920\right){x}+2686a-11488$ |
| 4.1-a4 |
4.1-a |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{28} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 7$ |
3B, 7B.6.1 |
$1$ |
\( 1 \) |
$1$ |
$7.662151750$ |
1.014876791 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 214 a + 705\) , \( -2687 a - 8802\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(214a+705\right){x}-2687a-8802$ |
| 4.1-b1 |
4.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$37.29397324$ |
2.469853714 |
\( -\frac{20297286875}{64} a + \frac{43383068421}{32} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 91 a - 386\) , \( -817 a + 3486\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(91a-386\right){x}-817a+3486$ |
| 4.1-b2 |
4.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{3} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$37.29397324$ |
2.469853714 |
\( -\frac{489}{4} a + 1841 \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( a - 1\) , \( -7\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(a-1\right){x}-7$ |
| 4.1-b3 |
4.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{3} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$37.29397324$ |
2.469853714 |
\( \frac{489}{4} a + \frac{6875}{4} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -2 a\) , \( -a - 7\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}-2a{x}-a-7$ |
| 4.1-b4 |
4.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$37.29397324$ |
2.469853714 |
\( \frac{20297286875}{64} a + \frac{66468849967}{64} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -92 a - 295\) , \( 816 a + 2669\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-92a-295\right){x}+816a+2669$ |
| 4.1-c1 |
4.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$3.156724381$ |
0.209059179 |
\( -\frac{20297286875}{64} a + \frac{43383068421}{32} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( -2960 a - 9690\) , \( -168414 a - 551544\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2960a-9690\right){x}-168414a-551544$ |
| 4.1-c2 |
4.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{3} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$28.41051943$ |
0.209059179 |
\( -\frac{489}{4} a + 1841 \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 150 a + 495\) , \( -1331 a - 4361\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(150a+495\right){x}-1331a-4361$ |
| 4.1-c3 |
4.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{3} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$28.41051943$ |
0.209059179 |
\( \frac{489}{4} a + \frac{6875}{4} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -149 a + 645\) , \( 1181 a - 5047\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-149a+645\right){x}+1181a-5047$ |
| 4.1-c4 |
4.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$3.156724381$ |
0.209059179 |
\( \frac{20297286875}{64} a + \frac{66468849967}{64} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( 2961 a - 12650\) , \( 171374 a - 732608\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2961a-12650\right){x}+171374a-732608$ |
| 4.1-d1 |
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 |
\( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -642629 a - 2104558\) , \( -545795629 a - 1787435506\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-642629a-2104558\right){x}-545795629a-1787435506$ |
| 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 |
\( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 642628 a - 2747187\) , \( 545795628 a - 2333231135\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(642628a-2747187\right){x}+545795628a-2333231135$ |
| 4.1-d3 |
4.1-d |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{28} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 7$ |
3B, 7B.1.1 |
$1$ |
\( 3 \cdot 7^{2} \) |
$1$ |
$3.815467665$ |
1.516113114 |
\( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -239 a - 778\) , \( -6021 a - 19722\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-239a-778\right){x}-6021a-19722$ |
| 4.1-d4 |
4.1-d |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{28} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 7$ |
3B, 7B.1.1 |
$1$ |
\( 3 \cdot 7^{2} \) |
$1$ |
$3.815467665$ |
1.516113114 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 238 a - 1017\) , \( 6020 a - 25743\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(238a-1017\right){x}+6020a-25743$ |
*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.
