| 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 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 238 a - 1017\) , \( 6020 a - 25743\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(238a-1017\right){x}+6020a-25743$ |
| 4.2-a1 |
4.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{8} \) |
$0.95409$ |
$(a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$19$ |
19Ns.4.1 |
$1$ |
\( 1 \) |
$0.228592103$ |
$17.69503190$ |
1.071531991 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( a + 5\) , \( -a - 5\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+5\right){x}-a-5$ |
| 4.2-a2 |
4.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{8} \) |
$0.95409$ |
$(a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$19$ |
19Ns.4.1 |
$1$ |
\( 3 \) |
$0.076197367$ |
$17.69503190$ |
1.071531991 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( a + 5\) , \( 17422 a - 74490\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+5\right){x}+17422a-74490$ |
| 4.3-a1 |
4.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.3 |
\( 2^{2} \) |
\( 2^{8} \) |
$0.95409$ |
$(a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$19$ |
19Ns.4.1 |
$1$ |
\( 1 \) |
$0.228592103$ |
$17.69503190$ |
1.071531991 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( a + 5\) , \( -10\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+5\right){x}-10$ |
| 4.3-a2 |
4.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.3 |
\( 2^{2} \) |
\( 2^{8} \) |
$0.95409$ |
$(a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$19$ |
19Ns.4.1 |
$1$ |
\( 3 \) |
$0.076197367$ |
$17.69503190$ |
1.071531991 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( a + 5\) , \( -17423 a - 57062\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+5\right){x}-17423a-57062$ |
| 8.3-a1 |
8.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( 2^{8} \) |
$1.13461$ |
$(a-4)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$5.977571122$ |
3.166994547 |
\( -7168 a - 23552 \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -a - 3\) , \( -a - 6\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(-a-3\right){x}-a-6$ |
| 8.3-b1 |
8.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( 2^{8} \) |
$1.13461$ |
$(a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.031315266$ |
$41.25238393$ |
0.684427934 |
\( -7168 a - 23552 \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( 706 a - 3012\) , \( -94983 a + 406037\bigr] \) |
${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(706a-3012\right){x}-94983a+406037$ |
| 8.4-a1 |
8.4-a |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( 2^{8} \) |
$1.13461$ |
$(a+3)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$5.977571122$ |
3.166994547 |
\( 7168 a - 30720 \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( a - 4\) , \( -7\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(a-4\right){x}-7$ |
| 8.4-b1 |
8.4-b |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( 2^{8} \) |
$1.13461$ |
$(a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.031315266$ |
$41.25238393$ |
0.684427934 |
\( 7168 a - 30720 \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( -706 a - 2306\) , \( 94982 a + 311054\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-706a-2306\right){x}+94982a+311054$ |
| 14.2-a1 |
14.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{3} \) |
$1.30499$ |
$(a-4), (-2a+9)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$4.972949964$ |
1.317366627 |
\( -\frac{545882409}{5488} a - \frac{254194875}{784} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -14093 a + 60285\) , \( 655027 a - 2800123\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-14093a+60285\right){x}+655027a-2800123$ |
| 14.2-a2 |
14.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{12} \cdot 7 \) |
$1.30499$ |
$(a-4), (-2a+9)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$4.972949964$ |
1.317366627 |
\( \frac{116429967}{28672} a - \frac{70335891}{4096} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( -a - 5\) , \( -5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-5\right){x}-5$ |
| 14.2-b1 |
14.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{3} \) |
$1.30499$ |
$(a-4), (-2a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.060818804$ |
$23.51278497$ |
1.515285641 |
\( -\frac{545882409}{5488} a - \frac{254194875}{784} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( -2 a\) , \( 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}-2a{x}+3$ |
| 14.2-b2 |
14.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{12} \cdot 7 \) |
$1.30499$ |
$(a-4), (-2a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.020272934$ |
$23.51278497$ |
1.515285641 |
\( \frac{116429967}{28672} a - \frac{70335891}{4096} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 12784 a + 41868\) , \( 4430171 a + 14508443\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(12784a+41868\right){x}+4430171a+14508443$ |
| 14.3-a1 |
14.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{12} \cdot 7 \) |
$1.30499$ |
$(a+3), (2a+7)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$4.972949964$ |
1.317366627 |
\( -\frac{116429967}{28672} a - \frac{187960635}{14336} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -a - 5\) , \( -a - 5\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-a-5\right){x}-a-5$ |
| 14.3-a2 |
14.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{3} \) |
$1.30499$ |
$(a+3), (2a+7)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$4.972949964$ |
1.317366627 |
\( \frac{545882409}{5488} a - \frac{1162623267}{2744} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 14101 a + 46176\) , \( -608844 a - 1993915\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(14101a+46176\right){x}-608844a-1993915$ |
| 14.3-b1 |
14.3-b |
$2$ |
$3$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{12} \cdot 7 \) |
$1.30499$ |
$(a+3), (2a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.020272934$ |
$23.51278497$ |
1.515285641 |
\( -\frac{116429967}{28672} a - \frac{187960635}{14336} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -12775 a + 54636\) , \( -4375534 a + 18705071\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-12775a+54636\right){x}-4375534a+18705071$ |
| 14.3-b2 |
14.3-b |
$2$ |
$3$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{3} \) |
$1.30499$ |
$(a+3), (2a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.060818804$ |
$23.51278497$ |
1.515285641 |
\( \frac{545882409}{5488} a - \frac{1162623267}{2744} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( a - 1\) , \( 3\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(a-1\right){x}+3$ |
| 16.1-a1 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2, 19$ |
2B, 19Ns.4.1 |
$1$ |
\( 3 \) |
$1$ |
$17.69503190$ |
1.757823174 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 5\) , \( 133 a - 574\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+5\right){x}+133a-574$ |
| 16.1-a2 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2, 19$ |
2B, 19Ns.4.1 |
$1$ |
\( 3 \) |
$1$ |
$17.69503190$ |
1.757823174 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 5\) , \( -133 a - 436\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+5\right){x}-133a-436$ |
| 16.1-a3 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{16} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$19$ |
19Ns.4.1 |
$1$ |
\( 3 \) |
$1$ |
$17.69503190$ |
1.757823174 |
\( 54000 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 201 a - 850\) , \( 3094 a - 13232\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(201a-850\right){x}+3094a-13232$ |
| 16.1-a4 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{16} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$19$ |
19Ns.4.1 |
$1$ |
\( 3 \) |
$1$ |
$17.69503190$ |
1.757823174 |
\( 54000 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -199 a - 650\) , \( -3294 a - 10788\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-199a-650\right){x}-3294a-10788$ |
| 16.2-a1 |
16.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.2 |
\( 2^{4} \) |
\( - 2^{15} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$22.49427591$ |
1.489719814 |
\( -\frac{925430099}{32} a + \frac{3956137073}{32} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 320 a + 1048\) , \( 469916 a + 1538936\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(320a+1048\right){x}+469916a+1538936$ |
| 16.2-a2 |
16.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.2 |
\( 2^{4} \) |
\( - 2^{21} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$22.49427591$ |
1.489719814 |
\( \frac{7754659}{1024} a + \frac{11013279}{1024} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 888 a - 3768\) , \( -30040 a + 128464\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(888a-3768\right){x}-30040a+128464$ |
| 16.2-b1 |
16.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.2 |
\( 2^{4} \) |
\( - 2^{15} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$4.340752186$ |
1.437366681 |
\( -\frac{925430099}{32} a + \frac{3956137073}{32} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 27 a - 113\) , \( 195 a - 833\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(27a-113\right){x}+195a-833$ |
| 16.2-b2 |
16.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.2 |
\( 2^{4} \) |
\( - 2^{21} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$4.340752186$ |
1.437366681 |
\( \frac{7754659}{1024} a + \frac{11013279}{1024} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -84 a - 276\) , \( -1037 a - 3397\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-84a-276\right){x}-1037a-3397$ |
| 16.2-c1 |
16.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.2 |
\( 2^{4} \) |
\( 2^{12} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$1$ |
$22.13736152$ |
2.932165164 |
\( \frac{4171}{2} a - \frac{14469}{2} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 4 a + 20\) , \( 2 a + 10\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(4a+20\right){x}+2a+10$ |
| 16.2-d1 |
16.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.2 |
\( 2^{4} \) |
\( 2^{12} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$1$ |
$7.567530696$ |
1.002343927 |
\( \frac{4171}{2} a - \frac{14469}{2} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 693 a - 2958\) , \( 21766 a - 93046\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(693a-2958\right){x}+21766a-93046$ |
| 16.3-a1 |
16.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.3 |
\( 2^{4} \) |
\( - 2^{21} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$22.49427591$ |
1.489719814 |
\( -\frac{7754659}{1024} a + \frac{9383969}{512} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -882 a - 2893\) , \( 27153 a + 88921\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-882a-2893\right){x}+27153a+88921$ |
| 16.3-a2 |
16.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.3 |
\( 2^{4} \) |
\( - 2^{15} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$22.49427591$ |
1.489719814 |
\( \frac{925430099}{32} a + \frac{1515353487}{16} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -320 a + 1368\) , \( -469916 a + 2008852\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-320a+1368\right){x}-469916a+2008852$ |
| 16.3-b1 |
16.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.3 |
\( 2^{4} \) |
\( - 2^{21} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$4.340752186$ |
1.437366681 |
\( -\frac{7754659}{1024} a + \frac{9383969}{512} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 84 a - 359\) , \( 953 a - 4074\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(84a-359\right){x}+953a-4074$ |
| 16.3-b2 |
16.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.3 |
\( 2^{4} \) |
\( - 2^{15} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$4.340752186$ |
1.437366681 |
\( \frac{925430099}{32} a + \frac{1515353487}{16} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -27 a - 86\) , \( -195 a - 638\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-27a-86\right){x}-195a-638$ |
| 16.3-c1 |
16.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{12} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$1$ |
$22.13736152$ |
2.932165164 |
\( -\frac{4171}{2} a - 5149 \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -4 a + 24\) , \( -2 a + 12\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-4a+24\right){x}-2a+12$ |
| 16.3-d1 |
16.3-d |
$1$ |
$1$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{12} \) |
$1.34929$ |
$(a-4), (a+3)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$1$ |
$7.567530696$ |
1.002343927 |
\( -\frac{4171}{2} a - 5149 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -693 a - 2265\) , \( -21766 a - 71280\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-693a-2265\right){x}-21766a-71280$ |
| 16.4-a1 |
16.4-a |
$2$ |
$19$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{12} \) |
$1.34929$ |
$(a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$3.249844398$ |
$2.014351525$ |
1.734164921 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 41678 a - 178170\) , \( 9007428 a - 38506016\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(41678a-178170\right){x}+9007428a-38506016$ |
| 16.4-a2 |
16.4-a |
$2$ |
$19$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{12} \) |
$1.34929$ |
$(a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.171044442$ |
$38.27267897$ |
1.734164921 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( -2 a - 10\) , \( 3 a + 11\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-2a-10\right){x}+3a+11$ |
*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.
