| 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 |
| 153.1-a1 |
153.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
153.1 |
\( 3^{2} \cdot 17 \) |
\( 3^{8} \cdot 17^{4} \) |
$2.89748$ |
$(3,a), (3,a+2), (17,a+8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$2.636651833$ |
0.571970089 |
\( -\frac{13377536}{70227} a - \frac{16928768}{23409} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -a + 3\) , \( 4 a - 22\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-a+3\right){x}+4a-22$ |
| 153.1-b1 |
153.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
153.1 |
\( 3^{2} \cdot 17 \) |
\( 3^{20} \cdot 17^{4} \) |
$2.89748$ |
$(3,a), (3,a+2), (17,a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \cdot 3 \cdot 5 \) |
$0.175283747$ |
$2.636651833$ |
6.015423658 |
\( -\frac{13377536}{70227} a - \frac{16928768}{23409} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( -75 a + 390\) , \( 4126 a - 21083\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(-75a+390\right){x}+4126a-21083$ |
| 153.1-c1 |
153.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
153.1 |
\( 3^{2} \cdot 17 \) |
\( 3^{2} \cdot 17^{6} \) |
$2.89748$ |
$(3,a), (3,a+2), (17,a+8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$0.739701511$ |
1.444174087 |
\( -\frac{23100424192}{14739} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -59\) , \( -196\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-59{x}-196$ |
| 153.1-c2 |
153.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
153.1 |
\( 3^{2} \cdot 17 \) |
\( 3^{6} \cdot 17^{2} \) |
$2.89748$ |
$(3,a), (3,a+2), (17,a+8)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$6.657313600$ |
1.444174087 |
\( \frac{32768}{459} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 1\) , \( -1\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+{x}-1$ |
| 153.1-d1 |
153.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
153.1 |
\( 3^{2} \cdot 17 \) |
\( 3^{14} \cdot 17^{6} \) |
$2.89748$ |
$(3,a), (3,a+2), (17,a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$2.035913469$ |
$0.739701511$ |
1.960142318 |
\( -\frac{23100424192}{14739} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 653 a - 3375\) , \( 20622 a - 105423\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(653a-3375\right){x}+20622a-105423$ |
| 153.1-d2 |
153.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
153.1 |
\( 3^{2} \cdot 17 \) |
\( 3^{18} \cdot 17^{2} \) |
$2.89748$ |
$(3,a), (3,a+2), (17,a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.678637823$ |
$6.657313600$ |
1.960142318 |
\( \frac{32768}{459} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -7 a + 45\) , \( 102 a - 528\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(-7a+45\right){x}+102a-528$ |
| 153.1-e1 |
153.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
153.1 |
\( 3^{2} \cdot 17 \) |
\( 3^{20} \cdot 17^{4} \) |
$2.89748$ |
$(3,a), (3,a+2), (17,a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \cdot 3 \cdot 5 \) |
$0.175283747$ |
$2.636651833$ |
6.015423658 |
\( \frac{13377536}{70227} a - \frac{64163840}{70227} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 28 a - 129\) , \( 196 a - 1013\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(28a-129\right){x}+196a-1013$ |
| 153.1-f1 |
153.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
153.1 |
\( 3^{2} \cdot 17 \) |
\( 3^{8} \cdot 17^{4} \) |
$2.89748$ |
$(3,a), (3,a+2), (17,a+8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$2.636651833$ |
0.571970089 |
\( \frac{13377536}{70227} a - \frac{64163840}{70227} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( a + 2\) , \( -4 a - 18\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(a+2\right){x}-4a-18$ |
*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.
