| 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 |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$13.09573$ |
$(1/2a^2+a-1/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$365.8824173$ |
1.642860551 |
\( \frac{8025637785355}{81} a^{3} + \frac{46061580093845}{162} a^{2} - \frac{30209462949730}{81} a - \frac{57788918959565}{54} \) |
\( \bigl[\frac{1}{2} a^{2} - \frac{5}{2}\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + \frac{5}{2} a - \frac{9}{2}\) , \( a\) , \( -\frac{37}{2} a^{3} + 35 a^{2} + \frac{301}{2} a - 289\) , \( -102 a^{3} + \frac{403}{2} a^{2} + 846 a - \frac{3309}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}-\frac{5}{2}\right){x}{y}+a{y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+\frac{5}{2}a-\frac{9}{2}\right){x}^{2}+\left(-\frac{37}{2}a^{3}+35a^{2}+\frac{301}{2}a-289\right){x}-102a^{3}+\frac{403}{2}a^{2}+846a-\frac{3309}{2}$ |
| 9.1-a2 |
9.1-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$13.09573$ |
$(1/2a^2+a-1/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$365.8824173$ |
1.642860551 |
\( -\frac{29105995}{9} a^{3} + \frac{173474645}{18} a^{2} + \frac{36523615}{3} a - \frac{653017795}{18} \) |
\( \bigl[\frac{1}{2} a^{2} - \frac{5}{2}\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + \frac{5}{2} a - \frac{9}{2}\) , \( a\) , \( -a^{3} + \frac{5}{2} a^{2} + 8 a - \frac{33}{2}\) , \( -\frac{1}{2} a^{3} + \frac{11}{2} a - 8\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}-\frac{5}{2}\right){x}{y}+a{y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+\frac{5}{2}a-\frac{9}{2}\right){x}^{2}+\left(-a^{3}+\frac{5}{2}a^{2}+8a-\frac{33}{2}\right){x}-\frac{1}{2}a^{3}+\frac{11}{2}a-8$ |
| 9.1-a3 |
9.1-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{24} \) |
$13.09573$ |
$(1/2a^2+a-1/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$365.8824173$ |
1.642860551 |
\( -\frac{18006809585}{1062882} a^{3} + \frac{25720340240}{531441} a^{2} + \frac{70788991685}{1062882} a - \frac{30982326505}{177147} \) |
\( \bigl[\frac{1}{2} a^{2} + a - \frac{7}{2}\) , \( -\frac{1}{2} a^{3} + \frac{9}{2} a\) , \( \frac{1}{2} a^{3} - \frac{5}{2} a + 1\) , \( 7 a^{3} + \frac{13}{2} a^{2} - 53 a - \frac{143}{2}\) , \( 31 a^{3} + \frac{139}{2} a^{2} - 265 a - \frac{1101}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{5}{2}a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{9}{2}a\right){x}^{2}+\left(7a^{3}+\frac{13}{2}a^{2}-53a-\frac{143}{2}\right){x}+31a^{3}+\frac{139}{2}a^{2}-265a-\frac{1101}{2}$ |
| 9.1-a4 |
9.1-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{12} \) |
$13.09573$ |
$(1/2a^2+a-1/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$365.8824173$ |
1.642860551 |
\( \frac{51065}{243} a^{3} + \frac{776305}{1458} a^{2} - \frac{876295}{729} a - \frac{1624405}{1458} \) |
\( \bigl[\frac{1}{2} a^{2} + a - \frac{7}{2}\) , \( -\frac{1}{2} a^{3} + \frac{9}{2} a\) , \( \frac{1}{2} a^{3} - \frac{5}{2} a + 1\) , \( -\frac{1}{2} a^{3} - \frac{7}{2} a^{2} + \frac{9}{2} a + \frac{47}{2}\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{9}{2} a - \frac{17}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{5}{2}a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{9}{2}a\right){x}^{2}+\left(-\frac{1}{2}a^{3}-\frac{7}{2}a^{2}+\frac{9}{2}a+\frac{47}{2}\right){x}+\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{9}{2}a-\frac{17}{2}$ |
| 9.1-b1 |
9.1-b |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$13.09573$ |
$(1/2a^2+a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.153200176$ |
$525.2209558$ |
2.890350157 |
\( \frac{8025637785355}{81} a^{3} + \frac{46061580093845}{162} a^{2} - \frac{30209462949730}{81} a - \frac{57788918959565}{54} \) |
\( \bigl[a\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{9}{2} a + \frac{5}{2}\) , \( 0\) , \( -8 a^{3} + 7 a^{2} + 64 a - 78\) , \( \frac{79}{2} a^{3} - 72 a^{2} - \frac{631}{2} a + 604\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{9}{2}a+\frac{5}{2}\right){x}^{2}+\left(-8a^{3}+7a^{2}+64a-78\right){x}+\frac{79}{2}a^{3}-72a^{2}-\frac{631}{2}a+604$ |
| 9.1-b2 |
9.1-b |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$13.09573$ |
$(1/2a^2+a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.306400352$ |
$525.2209558$ |
2.890350157 |
\( -\frac{29105995}{9} a^{3} + \frac{173474645}{18} a^{2} + \frac{36523615}{3} a - \frac{653017795}{18} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{7}{2} a\) , \( -\frac{1}{2} a^{2} - a + \frac{7}{2}\) , \( 1\) , \( -\frac{5}{2} a^{3} + \frac{9}{2} a^{2} + \frac{41}{2} a - \frac{71}{2}\) , \( \frac{7}{2} a^{3} - 7 a^{2} - \frac{57}{2} a + 56\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{7}{2}a\right){x}{y}+{y}={x}^{3}+\left(-\frac{1}{2}a^{2}-a+\frac{7}{2}\right){x}^{2}+\left(-\frac{5}{2}a^{3}+\frac{9}{2}a^{2}+\frac{41}{2}a-\frac{71}{2}\right){x}+\frac{7}{2}a^{3}-7a^{2}-\frac{57}{2}a+56$ |
| 9.1-b3 |
9.1-b |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{24} \) |
$13.09573$ |
$(1/2a^2+a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.051066725$ |
$525.2209558$ |
2.890350157 |
\( -\frac{18006809585}{1062882} a^{3} + \frac{25720340240}{531441} a^{2} + \frac{70788991685}{1062882} a - \frac{30982326505}{177147} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{7}{2}\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( \frac{13}{2} a^{3} - a^{2} - \frac{75}{2} a - 15\) , \( -\frac{21}{2} a^{3} + 10 a^{2} + \frac{129}{2} a + 9\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+1\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+1\right){x}^{2}+\left(\frac{13}{2}a^{3}-a^{2}-\frac{75}{2}a-15\right){x}-\frac{21}{2}a^{3}+10a^{2}+\frac{129}{2}a+9$ |
| 9.1-b4 |
9.1-b |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{12} \) |
$13.09573$ |
$(1/2a^2+a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.102133450$ |
$525.2209558$ |
2.890350157 |
\( \frac{51065}{243} a^{3} + \frac{776305}{1458} a^{2} - \frac{876295}{729} a - \frac{1624405}{1458} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{7}{2}\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( \frac{3}{2} a^{3} + \frac{3}{2} a^{2} - \frac{15}{2} a - \frac{5}{2}\) , \( a^{3} + \frac{5}{2} a^{2} - 2 a - \frac{15}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+1\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+1\right){x}^{2}+\left(\frac{3}{2}a^{3}+\frac{3}{2}a^{2}-\frac{15}{2}a-\frac{5}{2}\right){x}+a^{3}+\frac{5}{2}a^{2}-2a-\frac{15}{2}$ |
*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.
