| 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 |
| 20956.6-a1 |
20956.6-a |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20956.6 |
\( 2^{2} \cdot 13^{2} \cdot 31 \) |
\( 2^{2} \cdot 13^{3} \cdot 31 \) |
$1.86220$ |
$(4a-3), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.127482162$ |
$4.884849081$ |
2.876271852 |
\( -\frac{11259}{31} a + \frac{38907}{62} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( a - 2\) , \( -a - 1\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(a-2\right){x}-a-1$ |
| 20956.6-b1 |
20956.6-b |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20956.6 |
\( 2^{2} \cdot 13^{2} \cdot 31 \) |
\( 2^{2} \cdot 13^{9} \cdot 31 \) |
$1.86220$ |
$(4a-3), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.485799319$ |
$1.354813371$ |
3.039945067 |
\( -\frac{11259}{31} a + \frac{38907}{62} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -7 a + 21\) , \( 3 a - 53\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-7a+21\right){x}+3a-53$ |
| 20956.6-c1 |
20956.6-c |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20956.6 |
\( 2^{2} \cdot 13^{2} \cdot 31 \) |
\( 2^{50} \cdot 13^{6} \cdot 31 \) |
$1.86220$ |
$(4a-3), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 5^{2} \) |
$1$ |
$0.102184592$ |
2.949815085 |
\( \frac{936087656892551}{1040187392} a - \frac{833285178768245}{1040187392} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 13496 a - 15653\) , \( 772288 a - 509794\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(13496a-15653\right){x}+772288a-509794$ |
| 20956.6-c2 |
20956.6-c |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20956.6 |
\( 2^{2} \cdot 13^{2} \cdot 31 \) |
\( 2^{2} \cdot 13^{6} \cdot 31 \) |
$1.86220$ |
$(4a-3), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$2.554614800$ |
2.949815085 |
\( \frac{24551}{62} a + \frac{66955}{62} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -4 a - 3\) , \( 8 a - 4\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4a-3\right){x}+8a-4$ |
| 20956.6-c3 |
20956.6-c |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20956.6 |
\( 2^{2} \cdot 13^{2} \cdot 31 \) |
\( 2^{10} \cdot 13^{6} \cdot 31^{5} \) |
$1.86220$ |
$(4a-3), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5Cs.4.1 |
$1$ |
\( 5 \) |
$1$ |
$0.510922960$ |
2.949815085 |
\( -\frac{511363962461}{916132832} a + \frac{764718499383}{458066416} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -149 a + 187\) , \( -322 a + 510\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-149a+187\right){x}-322a+510$ |
| 20956.6-d1 |
20956.6-d |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20956.6 |
\( 2^{2} \cdot 13^{2} \cdot 31 \) |
\( 2^{18} \cdot 13^{7} \cdot 31 \) |
$1.86220$ |
$(4a-3), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{2} \) |
$1.987439123$ |
$0.196533883$ |
3.608200201 |
\( \frac{692551214999273}{103168} a - \frac{1147892738295865}{206336} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 12367 a - 4587\) , \( -268995 a - 229235\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(12367a-4587\right){x}-268995a-229235$ |
| 20956.6-d2 |
20956.6-d |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20956.6 |
\( 2^{2} \cdot 13^{2} \cdot 31 \) |
\( 2^{2} \cdot 13^{7} \cdot 31 \) |
$1.86220$ |
$(4a-3), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{2} \) |
$0.220826569$ |
$1.768804953$ |
3.608200201 |
\( \frac{21320199}{806} a - \frac{2265127}{403} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -23 a - 7\) , \( -51 a + 7\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-23a-7\right){x}-51a+7$ |
| 20956.6-d3 |
20956.6-d |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20956.6 |
\( 2^{2} \cdot 13^{2} \cdot 31 \) |
\( 2^{6} \cdot 13^{9} \cdot 31^{3} \) |
$1.86220$ |
$(4a-3), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs[2] |
$1$ |
\( 2^{2} \) |
$0.662479707$ |
$0.589601651$ |
3.608200201 |
\( \frac{312668878725}{261803308} a + \frac{892296547723}{523606616} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 167 a - 52\) , \( -409 a - 95\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(167a-52\right){x}-409a-95$ |
| 20956.6-d4 |
20956.6-d |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20956.6 |
\( 2^{2} \cdot 13^{2} \cdot 31 \) |
\( 2^{2} \cdot 13^{7} \cdot 31^{9} \) |
$1.86220$ |
$(4a-3), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{2} \) |
$1.987439123$ |
$0.196533883$ |
3.608200201 |
\( -\frac{3946979270626725529}{687430176177446} a + \frac{2492130373024223175}{687430176177446} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -1783 a + 988\) , \( 20747 a - 26115\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-1783a+988\right){x}+20747a-26115$ |
| 20956.6-d5 |
20956.6-d |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20956.6 |
\( 2^{2} \cdot 13^{2} \cdot 31 \) |
\( 2^{2} \cdot 13^{15} \cdot 31 \) |
$1.86220$ |
$(4a-3), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{2} \) |
$1.987439123$ |
$0.196533883$ |
3.608200201 |
\( -\frac{2293457084040902583}{328739480563} a + \frac{3222747199562278231}{657478961126} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 5307 a - 202\) , \( 7099 a + 142097\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(5307a-202\right){x}+7099a+142097$ |
*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.
