| 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 |
| 9.1-a1 |
9.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \cdot 5^{12} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
|
|
$9$ |
\( 1 \) |
$1$ |
$9.266583183$ |
6.925930624 |
\( -\frac{4096}{3} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -400 a - 2208\) , \( -15950 a - 88057\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-400a-2208\right){x}-15950a-88057$ |
| 9.1-b1 |
9.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 2^{12} \cdot 3^{4} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$35.21805767$ |
2.924700499 |
\( -\frac{82475026}{9} a + \frac{537804017}{9} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 1512 a - 9872\) , \( -71797 a + 468168\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1512a-9872\right){x}-71797a+468168$ |
| 9.1-b2 |
9.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{20} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$2.201128604$ |
2.924700499 |
\( -\frac{91151604355202}{43046721} a + \frac{66044208624673}{4782969} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 5147 a - 33563\) , \( 493610 a - 3218731\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(5147a-33563\right){x}+493610a-3218731$ |
| 9.1-b3 |
9.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{16} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$8.804514419$ |
2.924700499 |
\( \frac{38272753}{6561} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 337 a - 2198\) , \( 6960 a - 45385\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(337a-2198\right){x}+6960a-45385$ |
| 9.1-b4 |
9.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$35.21805767$ |
2.924700499 |
\( \frac{912673}{81} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 97 a - 633\) , \( -1166 a + 7603\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(97a-633\right){x}-1166a+7603$ |
| 9.1-b5 |
9.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{20} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$2.201128604$ |
2.924700499 |
\( \frac{91151604355202}{43046721} a + \frac{503246273266855}{43046721} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -633 a + 4127\) , \( 39414 a - 257011\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-633a+4127\right){x}+39414a-257011$ |
| 9.1-b6 |
9.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 2^{12} \cdot 3^{4} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$35.21805767$ |
2.924700499 |
\( \frac{82475026}{9} a + \frac{455328991}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -105 a + 723\) , \( -5658 a + 36985\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-105a+723\right){x}-5658a+36985$ |
| 9.1-c1 |
9.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 2^{12} \cdot 3^{4} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.394024136$ |
$35.21805767$ |
1.152402589 |
\( -\frac{82475026}{9} a + \frac{537804017}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 22 a - 95\) , \( -57 a + 488\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(22a-95\right){x}-57a+488$ |
| 9.1-c2 |
9.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{20} \cdot 5^{12} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.576096547$ |
$2.201128604$ |
1.152402589 |
\( -\frac{91151604355202}{43046721} a + \frac{66044208624673}{4782969} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 128675 a - 839063\) , \( 61701250 a - 402341344\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(128675a-839063\right){x}+61701250a-402341344$ |
| 9.1-c3 |
9.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{16} \cdot 5^{12} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.576096547$ |
$8.804514419$ |
1.152402589 |
\( \frac{38272753}{6561} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 8425 a - 54938\) , \( 870000 a - 5673094\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(8425a-54938\right){x}+870000a-5673094$ |
| 9.1-c4 |
9.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \cdot 5^{12} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
$2$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.576096547$ |
$35.21805767$ |
1.152402589 |
\( \frac{912673}{81} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 2425 a - 15813\) , \( -145750 a + 950406\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(2425a-15813\right){x}-145750a+950406$ |
| 9.1-c5 |
9.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{20} \cdot 5^{12} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.576096547$ |
$2.201128604$ |
1.152402589 |
\( \frac{91151604355202}{43046721} a + \frac{503246273266855}{43046721} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -15825 a + 103187\) , \( 4926750 a - 32126344\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-15825a+103187\right){x}+4926750a-32126344$ |
| 9.1-c6 |
9.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 2^{12} \cdot 3^{4} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.394024136$ |
$35.21805767$ |
1.152402589 |
\( \frac{82475026}{9} a + \frac{455328991}{9} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -22 a - 72\) , \( 79 a + 504\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-22a-72\right){x}+79a+504$ |
| 9.1-d1 |
9.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{145}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$1.86373$ |
$(3,a), (3,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$9.266583183$ |
0.769547847 |
\( -\frac{4096}{3} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 16 a - 104\) , \( 134 a - 874\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(16a-104\right){x}+134a-874$ |
*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.
