| 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-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 |
\( \frac{10947981839295713687464790827586754793436}{27} a^{4} - \frac{16910116865672541902732883385121893663963}{27} a^{3} - \frac{56478851274393237585283385062065675883994}{27} a^{2} + \frac{63601626922138511052720584288730205307096}{27} a + \frac{20103252580374097231986758925581632058597}{27} \) |
\( \bigl[-2 a^{4} + 3 a^{3} + 11 a^{2} - 10 a - 6\) , \( a^{3} - 6 a - 2\) , \( a^{3} - 5 a\) , \( 1135 a^{4} - 1723 a^{3} - 5889 a^{2} + 6565 a + 1790\) , \( 26992 a^{4} - 41489 a^{3} - 139745 a^{2} + 157332 a + 47655\bigr] \) |
${y}^2+\left(-2a^{4}+3a^{3}+11a^{2}-10a-6\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(a^{3}-6a-2\right){x}^{2}+\left(1135a^{4}-1723a^{3}-5889a^{2}+6565a+1790\right){x}+26992a^{4}-41489a^{3}-139745a^{2}+157332a+47655$ |
| 9.4-a4 |
9.4-a |
$4$ |
$21$ |
5.5.161121.1 |
$5$ |
$[5, 0]$ |
9.4 |
\( 3^{2} \) |
\( - 3^{9} \) |
$44.68265$ |
$(a^4-a^3-6a^2+3a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 7$ |
3B, 7B.6.3 |
$49$ |
\( 2 \) |
$1$ |
$14.70074920$ |
3.58913232 |
\( \frac{10947981839295713687464790827586754793436}{27} a^{4} - \frac{16910116865672541902732883385121893663963}{27} a^{3} - \frac{56478851274393237585283385062065675883994}{27} a^{2} + \frac{63601626922138511052720584288730205307096}{27} a + \frac{20103252580374097231986758925581632058597}{27} \) |
\( \bigl[-2 a^{4} + 3 a^{3} + 11 a^{2} - 10 a - 5\) , \( 2 a^{4} - 2 a^{3} - 11 a^{2} + 7 a + 4\) , \( 2 a^{4} - 2 a^{3} - 11 a^{2} + 7 a + 5\) , \( 11836 a^{4} - 18822 a^{3} - 60636 a^{2} + 70987 a + 19305\) , \( -902294 a^{4} + 1396419 a^{3} + 4636765 a^{2} - 5265527 a - 1600472\bigr] \) |
${y}^2+\left(-2a^{4}+3a^{3}+11a^{2}-10a-5\right){x}{y}+\left(2a^{4}-2a^{3}-11a^{2}+7a+5\right){y}={x}^{3}+\left(2a^{4}-2a^{3}-11a^{2}+7a+4\right){x}^{2}+\left(11836a^{4}-18822a^{3}-60636a^{2}+70987a+19305\right){x}-902294a^{4}+1396419a^{3}+4636765a^{2}-5265527a-1600472$ |
| 27.1-c4 |
27.1-c |
$4$ |
$21$ |
5.5.161121.1 |
$5$ |
$[5, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{9} \) |
$49.87134$ |
$(a^4-a^3-6a^2+3a+4), (-2a^4+3a^3+10a^2-10a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 7$ |
3B, 7B.6.3 |
$49$ |
\( 1 \) |
$1$ |
$8.185689570$ |
0.999252576 |
\( \frac{10947981839295713687464790827586754793436}{27} a^{4} - \frac{16910116865672541902732883385121893663963}{27} a^{3} - \frac{56478851274393237585283385062065675883994}{27} a^{2} + \frac{63601626922138511052720584288730205307096}{27} a + \frac{20103252580374097231986758925581632058597}{27} \) |
\( \bigl[-a^{4} + 2 a^{3} + 6 a^{2} - 8 a - 4\) , \( a^{4} - a^{3} - 5 a^{2} + 3 a + 1\) , \( -a^{4} + 2 a^{3} + 6 a^{2} - 8 a - 4\) , \( 35045 a^{4} - 54087 a^{3} - 180906 a^{2} + 203578 a + 64285\) , \( -4028726 a^{4} + 6222916 a^{3} + 20782036 a^{2} - 23402811 a - 7396955\bigr] \) |
${y}^2+\left(-a^{4}+2a^{3}+6a^{2}-8a-4\right){x}{y}+\left(-a^{4}+2a^{3}+6a^{2}-8a-4\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+3a+1\right){x}^{2}+\left(35045a^{4}-54087a^{3}-180906a^{2}+203578a+64285\right){x}-4028726a^{4}+6222916a^{3}+20782036a^{2}-23402811a-7396955$ |
| 81.6-g4 |
81.6-g |
$4$ |
$21$ |
5.5.161121.1 |
$5$ |
$[5, 0]$ |
81.6 |
\( 3^{4} \) |
\( - 3^{15} \) |
$55.66255$ |
$(a^4-a^3-6a^2+3a+4), (-2a^4+3a^3+10a^2-10a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 7$ |
3B.1.2, 7B.6.3 |
$49$ |
\( 2^{2} \) |
$6.878380989$ |
$0.318764579$ |
5.35311146 |
\( \frac{10947981839295713687464790827586754793436}{27} a^{4} - \frac{16910116865672541902732883385121893663963}{27} a^{3} - \frac{56478851274393237585283385062065675883994}{27} a^{2} + \frac{63601626922138511052720584288730205307096}{27} a + \frac{20103252580374097231986758925581632058597}{27} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( 2 a^{4} - 3 a^{3} - 11 a^{2} + 12 a + 5\) , \( -2 a^{4} + 3 a^{3} + 11 a^{2} - 10 a - 6\) , \( -1612 a^{4} + 859 a^{3} + 7034 a^{2} - 6057 a - 5218\) , \( -78263 a^{4} - 26450 a^{3} + 294726 a^{2} - 22700 a - 97502\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(-2a^{4}+3a^{3}+11a^{2}-10a-6\right){y}={x}^{3}+\left(2a^{4}-3a^{3}-11a^{2}+12a+5\right){x}^{2}+\left(-1612a^{4}+859a^{3}+7034a^{2}-6057a-5218\right){x}-78263a^{4}-26450a^{3}+294726a^{2}-22700a-97502$ |
*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.
