| 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 |
$4$ |
$6$ |
4.4.17069.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$15.36466$ |
$(a), (-a^2+2a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$0.434077405$ |
$328.6757440$ |
3.276061535 |
\( \frac{16679643524}{27} a^{3} - \frac{5013645460}{3} a^{2} - \frac{56448461584}{27} a + \frac{9792839605}{9} \) |
\( \bigl[a^{3} - a^{2} - 7 a - 3\) , \( -a^{3} + a^{2} + 7 a + 5\) , \( a^{3} - a^{2} - 6 a - 3\) , \( -6 a^{3} - a^{2} + 58 a + 55\) , \( -22 a^{3} + 24 a^{2} + 154 a + 105\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-7a-3\right){x}{y}+\left(a^{3}-a^{2}-6a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+7a+5\right){x}^{2}+\left(-6a^{3}-a^{2}+58a+55\right){x}-22a^{3}+24a^{2}+154a+105$ |
| 9.1-a2 |
9.1-a |
$4$ |
$6$ |
4.4.17069.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{8} \) |
$15.36466$ |
$(a), (-a^2+2a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.217038702$ |
$164.3378720$ |
3.276061535 |
\( -\frac{1477818427379603115196}{729} a^{3} + \frac{445055063225038905106}{81} a^{2} + \frac{4971503066625305022608}{729} a - \frac{864013551541148468249}{243} \) |
\( \bigl[a^{3} - a^{2} - 7 a - 3\) , \( -a^{3} + a^{2} + 7 a + 5\) , \( a^{3} - a^{2} - 6 a - 3\) , \( 44 a^{3} - 136 a^{2} - 112 a + 140\) , \( -340 a^{3} + 890 a^{2} + 1215 a - 470\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-7a-3\right){x}{y}+\left(a^{3}-a^{2}-6a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+7a+5\right){x}^{2}+\left(44a^{3}-136a^{2}-112a+140\right){x}-340a^{3}+890a^{2}+1215a-470$ |
| 9.1-a3 |
9.1-a |
$4$ |
$6$ |
4.4.17069.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{8} \) |
$15.36466$ |
$(a), (-a^2+2a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.072346234$ |
$1479.040848$ |
3.276061535 |
\( \frac{7509168023}{729} a^{3} + \frac{20859415606}{729} a^{2} + \frac{8989900874}{729} a - \frac{6451688741}{729} \) |
\( \bigl[a^{3} - 2 a^{2} - 5 a\) , \( a^{3} - 2 a^{2} - 6 a - 1\) , \( 0\) , \( 20 a^{3} - 44 a^{2} - 108 a + 44\) , \( 37 a^{3} - 83 a^{2} - 190 a + 103\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-5a\right){x}{y}={x}^{3}+\left(a^{3}-2a^{2}-6a-1\right){x}^{2}+\left(20a^{3}-44a^{2}-108a+44\right){x}+37a^{3}-83a^{2}-190a+103$ |
| 9.1-a4 |
9.1-a |
$4$ |
$6$ |
4.4.17069.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$15.36466$ |
$(a), (-a^2+2a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$0.144692468$ |
$2958.081696$ |
3.276061535 |
\( \frac{30713}{27} a^{3} - \frac{18341}{27} a^{2} - \frac{94933}{27} a + \frac{21478}{27} \) |
\( \bigl[1\) , \( -a + 1\) , \( a^{3} - a^{2} - 7 a - 4\) , \( 4 a^{3} - 2 a^{2} - 34 a - 30\) , \( -20 a^{3} + 12 a^{2} + 164 a + 146\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-7a-4\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(4a^{3}-2a^{2}-34a-30\right){x}-20a^{3}+12a^{2}+164a+146$ |
| 9.1-b1 |
9.1-b |
$4$ |
$6$ |
4.4.17069.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{8} \) |
$15.36466$ |
$(a), (-a^2+2a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.217038702$ |
$164.3378720$ |
3.276061535 |
\( -\frac{745831354853881921882}{729} a^{3} + \frac{441803995441619928202}{729} a^{2} + \frac{6146745844542120162782}{729} a + \frac{5488957417765283197501}{729} \) |
\( \bigl[a^{2} - 2 a - 3\) , \( a^{3} - 3 a^{2} - 4 a + 4\) , \( a^{3} - a^{2} - 6 a - 3\) , \( 26 a^{3} - 15 a^{2} - 217 a - 199\) , \( -128 a^{3} + 76 a^{2} + 1052 a + 935\bigr] \) |
${y}^2+\left(a^{2}-2a-3\right){x}{y}+\left(a^{3}-a^{2}-6a-3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-4a+4\right){x}^{2}+\left(26a^{3}-15a^{2}-217a-199\right){x}-128a^{3}+76a^{2}+1052a+935$ |
| 9.1-b2 |
9.1-b |
$4$ |
$6$ |
4.4.17069.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$15.36466$ |
$(a), (-a^2+2a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$0.434077405$ |
$328.6757440$ |
3.276061535 |
\( \frac{8338833284}{27} a^{3} - \frac{4914144476}{27} a^{2} - \frac{68643922456}{27} a - \frac{61306946069}{27} \) |
\( \bigl[a^{2} - 2 a - 3\) , \( a^{3} - 3 a^{2} - 4 a + 4\) , \( a^{3} - a^{2} - 6 a - 3\) , \( a^{3} - 12 a - 14\) , \( -a^{3} + 2 a^{2} + 3 a - 6\bigr] \) |
${y}^2+\left(a^{2}-2a-3\right){x}{y}+\left(a^{3}-a^{2}-6a-3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-4a+4\right){x}^{2}+\left(a^{3}-12a-14\right){x}-a^{3}+2a^{2}+3a-6$ |
| 9.1-b3 |
9.1-b |
$4$ |
$6$ |
4.4.17069.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{8} \) |
$15.36466$ |
$(a), (-a^2+2a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.072346234$ |
$1479.040848$ |
3.276061535 |
\( \frac{161678163968}{729} a^{3} - \frac{39914897732}{81} a^{2} - \frac{854926560829}{729} a + \frac{132953856256}{243} \) |
\( \bigl[a^{3} - 2 a^{2} - 5 a\) , \( a + 1\) , \( a^{3} - 2 a^{2} - 5 a\) , \( 4 a^{2} + 10 a - 5\) , \( 5 a^{3} + 3 a^{2} - 11 a + 3\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-5a\right){x}{y}+\left(a^{3}-2a^{2}-5a\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a^{2}+10a-5\right){x}+5a^{3}+3a^{2}-11a+3$ |
| 9.1-b4 |
9.1-b |
$4$ |
$6$ |
4.4.17069.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$15.36466$ |
$(a), (-a^2+2a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$0.144692468$ |
$2958.081696$ |
3.276061535 |
\( \frac{218600}{27} a^{3} - \frac{53365}{3} a^{2} - \frac{1151632}{27} a + \frac{170320}{9} \) |
\( \bigl[a^{3} - 2 a^{2} - 5 a\) , \( a + 1\) , \( a^{3} - 2 a^{2} - 5 a\) , \( -a^{2}\) , \( -a^{2} - a\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-5a\right){x}{y}+\left(a^{3}-2a^{2}-5a\right){y}={x}^{3}+\left(a+1\right){x}^{2}-a^{2}{x}-a^{2}-a$ |
*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.
