| 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 |
| 3.1-a1 |
3.1-a |
$2$ |
$3$ |
5.5.161121.1 |
$5$ |
$[5, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{3} \) |
$40.03379$ |
$(a^4-a^3-6a^2+3a+4)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$2494.450056$ |
2.07146445 |
\( -\frac{24629225242}{27} a^{4} + \frac{75813993140}{27} a^{3} - \frac{9784393355}{27} a^{2} - \frac{53552410306}{27} a - \frac{11850978778}{27} \) |
\( \bigl[-a^{4} + 2 a^{3} + 6 a^{2} - 7 a - 5\) , \( 2 a^{4} - 3 a^{3} - 11 a^{2} + 12 a + 6\) , \( 2 a^{4} - 2 a^{3} - 11 a^{2} + 6 a + 6\) , \( 4 a^{4} - 3 a^{3} - 21 a^{2} + 12 a + 16\) , \( -3 a^{4} - 3 a^{3} + 16 a^{2} + 22 a + 5\bigr] \) |
${y}^2+\left(-a^{4}+2a^{3}+6a^{2}-7a-5\right){x}{y}+\left(2a^{4}-2a^{3}-11a^{2}+6a+6\right){y}={x}^{3}+\left(2a^{4}-3a^{3}-11a^{2}+12a+6\right){x}^{2}+\left(4a^{4}-3a^{3}-21a^{2}+12a+16\right){x}-3a^{4}-3a^{3}+16a^{2}+22a+5$ |
| 3.1-a2 |
3.1-a |
$2$ |
$3$ |
5.5.161121.1 |
$5$ |
$[5, 0]$ |
3.1 |
\( 3 \) |
\( -3 \) |
$40.03379$ |
$(a^4-a^3-6a^2+3a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$92.38703912$ |
2.07146445 |
\( -\frac{371556534058387}{3} a^{4} - \frac{581110743937834}{3} a^{3} + \frac{739377005259757}{3} a^{2} + \frac{781085725804235}{3} a + \frac{144913403913515}{3} \) |
\( \bigl[-2 a^{4} + 3 a^{3} + 11 a^{2} - 11 a - 6\) , \( -2 a^{4} + 2 a^{3} + 11 a^{2} - 5 a - 6\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 8 a - 2\) , \( 3 a^{4} - 4 a^{3} - 16 a^{2} + 14 a + 7\) , \( 73 a^{4} + 21 a^{3} - 406 a^{2} - 307 a - 53\bigr] \) |
${y}^2+\left(-2a^{4}+3a^{3}+11a^{2}-11a-6\right){x}{y}+\left(-a^{4}+2a^{3}+5a^{2}-8a-2\right){y}={x}^{3}+\left(-2a^{4}+2a^{3}+11a^{2}-5a-6\right){x}^{2}+\left(3a^{4}-4a^{3}-16a^{2}+14a+7\right){x}+73a^{4}+21a^{3}-406a^{2}-307a-53$ |
| 3.1-b1 |
3.1-b |
$4$ |
$21$ |
5.5.161121.1 |
$5$ |
$[5, 0]$ |
3.1 |
\( 3 \) |
\( -3 \) |
$40.03379$ |
$(a^4-a^3-6a^2+3a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 7$ |
3B, 7B.1.3 |
$2401$ |
\( 1 \) |
$1$ |
$0.114743802$ |
0.686349502 |
\( -\frac{182694822791927563836971494}{3} a^{4} + \frac{562377064293000060272344751}{3} a^{3} - \frac{72581119082188519727955647}{3} a^{2} - \frac{397244058335937525189978733}{3} a - \frac{87908768509579034199639757}{3} \) |
\( \bigl[a^{4} - a^{3} - 5 a^{2} + 3 a + 2\) , \( 4 a^{4} - 5 a^{3} - 22 a^{2} + 17 a + 12\) , \( 2 a^{4} - 2 a^{3} - 11 a^{2} + 6 a + 5\) , \( 958 a^{4} - 1037 a^{3} - 5994 a^{2} + 4539 a + 3891\) , \( -11536 a^{4} + 19033 a^{3} + 50359 a^{2} - 45777 a - 36123\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-5a^{2}+3a+2\right){x}{y}+\left(2a^{4}-2a^{3}-11a^{2}+6a+5\right){y}={x}^{3}+\left(4a^{4}-5a^{3}-22a^{2}+17a+12\right){x}^{2}+\left(958a^{4}-1037a^{3}-5994a^{2}+4539a+3891\right){x}-11536a^{4}+19033a^{3}+50359a^{2}-45777a-36123$ |
| 3.1-b2 |
3.1-b |
$4$ |
$21$ |
5.5.161121.1 |
$5$ |
$[5, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{7} \) |
$40.03379$ |
$(a^4-a^3-6a^2+3a+4)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 7$ |
3B, 7B.1.1 |
$1$ |
\( 7 \) |
$1$ |
$1928.499085$ |
0.686349502 |
\( -\frac{167950302532}{2187} a^{4} + \frac{210167045039}{2187} a^{3} + \frac{950274978229}{2187} a^{2} - \frac{748701572032}{2187} a - \frac{640844374114}{2187} \) |
\( \bigl[-a^{4} + 2 a^{3} + 5 a^{2} - 7 a - 1\) , \( -a^{4} + a^{3} + 5 a^{2} - 3 a\) , \( -2 a^{4} + 3 a^{3} + 11 a^{2} - 11 a - 5\) , \( -11 a^{4} + 15 a^{3} + 58 a^{2} - 55 a - 24\) , \( -a^{2} + 4\bigr] \) |
${y}^2+\left(-a^{4}+2a^{3}+5a^{2}-7a-1\right){x}{y}+\left(-2a^{4}+3a^{3}+11a^{2}-11a-5\right){y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-3a\right){x}^{2}+\left(-11a^{4}+15a^{3}+58a^{2}-55a-24\right){x}-a^{2}+4$ |
| 3.1-b3 |
3.1-b |
$4$ |
$21$ |
5.5.161121.1 |
$5$ |
$[5, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{21} \) |
$40.03379$ |
$(a^4-a^3-6a^2+3a+4)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 7$ |
3B, 7B.1.1 |
$1$ |
\( 3 \cdot 7 \) |
$1$ |
$642.8330284$ |
0.686349502 |
\( \frac{1548900462329375}{10460353203} a^{4} - \frac{2393136318458227}{10460353203} a^{3} - \frac{7988065226618618}{10460353203} a^{2} + \frac{8997287030016374}{10460353203} a + \frac{2842653066201812}{10460353203} \) |
\( \bigl[-2 a^{4} + 3 a^{3} + 11 a^{2} - 10 a - 6\) , \( a^{3} - 6 a - 2\) , \( a^{3} - 5 a\) , \( -3 a^{3} + a^{2} + 15 a + 5\) , \( 3 a^{4} - 2 a^{3} - 16 a^{2} + 3 a + 3\bigr] \) |
${y}^2+\left(-2a^{4}+3a^{3}+11a^{2}-10a-6\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(a^{3}-6a-2\right){x}^{2}+\left(-3a^{3}+a^{2}+15a+5\right){x}+3a^{4}-2a^{3}-16a^{2}+3a+3$ |
| 3.1-b4 |
3.1-b |
$4$ |
$21$ |
5.5.161121.1 |
$5$ |
$[5, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{3} \) |
$40.03379$ |
$(a^4-a^3-6a^2+3a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 7$ |
3B, 7B.1.3 |
$2401$ |
\( 3 \) |
$1$ |
$0.038247934$ |
0.686349502 |
\( \frac{10947981839295713687464790827586754793436}{27} a^{4} - \frac{16910116865672541902732883385121893663963}{27} a^{3} - \frac{56478851274393237585283385062065675883994}{27} a^{2} + \frac{63601626922138511052720584288730205307096}{27} a + \frac{20103252580374097231986758925581632058597}{27} \) |
\( \bigl[-2 a^{4} + 3 a^{3} + 11 a^{2} - 10 a - 6\) , \( a^{3} - 6 a - 2\) , \( a^{3} - 5 a\) , \( 1135 a^{4} - 1723 a^{3} - 5889 a^{2} + 6565 a + 1790\) , \( 26992 a^{4} - 41489 a^{3} - 139745 a^{2} + 157332 a + 47655\bigr] \) |
${y}^2+\left(-2a^{4}+3a^{3}+11a^{2}-10a-6\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(a^{3}-6a-2\right){x}^{2}+\left(1135a^{4}-1723a^{3}-5889a^{2}+6565a+1790\right){x}+26992a^{4}-41489a^{3}-139745a^{2}+157332a+47655$ |
*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.
