| 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 |
| 361.1-a1 |
361.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( - 19^{3} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$4.901468693$ |
0.369462104 |
\( -\frac{686848000}{361} a + \frac{2276984000}{361} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -9 a - 22\) , \( -30 a - 97\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-9a-22\right){x}-30a-97$ |
| 361.1-a2 |
361.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( - 19^{9} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$4.901468693$ |
0.369462104 |
\( -\frac{7163840000}{47045881} a + \frac{84211448000}{47045881} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 56 a + 193\) , \( 278 a + 924\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(56a+193\right){x}+278a+924$ |
| 361.1-a3 |
361.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( - 19^{9} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$4.901468693$ |
0.369462104 |
\( \frac{7163840000}{47045881} a + \frac{84211448000}{47045881} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -57 a + 193\) , \( -278 a + 924\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-57a+193\right){x}-278a+924$ |
| 361.1-a4 |
361.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( - 19^{3} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$4.901468693$ |
0.369462104 |
\( \frac{686848000}{361} a + \frac{2276984000}{361} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 8 a - 22\) , \( 30 a - 97\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(8a-22\right){x}+30a-97$ |
| 361.1-b1 |
361.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( 19^{2} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$81$ |
\( 1 \) |
$1$ |
$0.205438503$ |
2.508652598 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -769\) , \( -8470\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-769{x}-8470$ |
| 361.1-b2 |
361.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( 19^{6} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$9$ |
\( 3^{2} \) |
$1$ |
$1.848946532$ |
2.508652598 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -9\) , \( -15\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-9{x}-15$ |
| 361.1-b3 |
361.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( 19^{2} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$9$ |
\( 1 \) |
$1$ |
$16.64051879$ |
2.508652598 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+{x}$ |
| 361.1-c1 |
361.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( 19^{2} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$17.03289160$ |
2.567805025 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -769\) , \( 8467\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}-769{x}+8467$ |
| 361.1-c2 |
361.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( 19^{6} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 1 \) |
$1$ |
$17.03289160$ |
2.567805025 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -9\) , \( 12\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}-9{x}+12$ |
| 361.1-c3 |
361.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( 19^{2} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$17.03289160$ |
2.567805025 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( 1\) , \( -3\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+{x}-3$ |
| 361.1-d1 |
361.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( - 19^{3} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$9$ |
\( 2 \) |
$1$ |
$33.19185138$ |
2.501929934 |
\( -\frac{686848000}{361} a + \frac{2276984000}{361} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -7 a - 27\) , \( 15 a + 47\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-7a-27\right){x}+15a+47$ |
| 361.1-d2 |
361.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( - 19^{9} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$3.687983487$ |
2.501929934 |
\( -\frac{7163840000}{47045881} a + \frac{84211448000}{47045881} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( 58 a + 188\) , \( -163 a - 544\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(58a+188\right){x}-163a-544$ |
| 361.1-d3 |
361.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( - 19^{9} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$3.687983487$ |
2.501929934 |
\( \frac{7163840000}{47045881} a + \frac{84211448000}{47045881} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( -59 a + 188\) , \( 163 a - 544\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-59a+188\right){x}+163a-544$ |
| 361.1-d4 |
361.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
361.1 |
\( 19^{2} \) |
\( - 19^{3} \) |
$2.58370$ |
$(2a-5), (-2a-5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$9$ |
\( 2 \) |
$1$ |
$33.19185138$ |
2.501929934 |
\( \frac{686848000}{361} a + \frac{2276984000}{361} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( 6 a - 27\) , \( -15 a + 47\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(6a-27\right){x}-15a+47$ |
*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.
