| 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 |
| 12337.5-a1 |
12337.5-a |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12337.5 |
\( 13^{2} \cdot 73 \) |
\( 13^{14} \cdot 73^{2} \) |
$1.63118$ |
$(4a-3), (-9a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.334723650$ |
0.773011159 |
\( -\frac{1179498260514472}{4347029012209} a - \frac{2798580139141481}{4347029012209} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 99 a - 361\) , \( 2208 a - 4530\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(99a-361\right){x}+2208a-4530$ |
| 12337.5-a2 |
12337.5-a |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12337.5 |
\( 13^{2} \cdot 73 \) |
\( 13^{10} \cdot 73 \) |
$1.63118$ |
$(4a-3), (-9a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$0.669447301$ |
0.773011159 |
\( \frac{1724307382288}{2084953} a + \frac{3070247407257}{2084953} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 134 a - 436\) , \( 1516 a - 3361\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(134a-436\right){x}+1516a-3361$ |
| 12337.5-b1 |
12337.5-b |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12337.5 |
\( 13^{2} \cdot 73 \) |
\( 13^{14} \cdot 73 \) |
$1.63118$ |
$(4a-3), (-9a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$0.400124902$ |
1.848097760 |
\( \frac{1879019850841272}{59548342633} a - \frac{2561535635057775}{59548342633} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 646 a - 442\) , \( 5316 a - 225\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(646a-442\right){x}+5316a-225$ |
| 12337.5-b2 |
12337.5-b |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12337.5 |
\( 13^{2} \cdot 73 \) |
\( 13^{10} \cdot 73^{2} \) |
$1.63118$ |
$(4a-3), (-9a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$0.800249804$ |
1.848097760 |
\( \frac{56772248112}{152201569} a - \frac{28005573663}{152201569} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 11 a - 47\) , \( 104 a + 200\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(11a-47\right){x}+104a+200$ |
| 12337.5-b3 |
12337.5-b |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12337.5 |
\( 13^{2} \cdot 73 \) |
\( 13^{8} \cdot 73 \) |
$1.63118$ |
$(4a-3), (-9a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.600499608$ |
1.848097760 |
\( -\frac{93760416}{12337} a + \frac{104503905}{12337} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -24 a + 28\) , \( 12 a + 35\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-24a+28\right){x}+12a+35$ |
| 12337.5-b4 |
12337.5-b |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12337.5 |
\( 13^{2} \cdot 73 \) |
\( 13^{8} \cdot 73^{4} \) |
$1.63118$ |
$(4a-3), (-9a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.400124902$ |
1.848097760 |
\( -\frac{2769791445149112}{4799302729} a + \frac{1752057461905119}{4799302729} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -64 a - 852\) , \( 1180 a + 9845\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-64a-852\right){x}+1180a+9845$ |
| 12337.5-c1 |
12337.5-c |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12337.5 |
\( 13^{2} \cdot 73 \) |
\( 13^{6} \cdot 73^{3} \) |
$1.63118$ |
$(4a-3), (-9a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$0.899261677$ |
3.115133830 |
\( \frac{60988685561}{389017} a - \frac{169775626841}{389017} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 180 a - 151\) , \( -1007 a + 266\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(180a-151\right){x}-1007a+266$ |
| 12337.5-c2 |
12337.5-c |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12337.5 |
\( 13^{2} \cdot 73 \) |
\( 13^{6} \cdot 73^{2} \) |
$1.63118$ |
$(4a-3), (-9a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$1.348892516$ |
3.115133830 |
\( -\frac{927841113}{5329} a - \frac{395933743}{5329} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -40 a + 79\) , \( -147 a - 80\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-40a+79\right){x}-147a-80$ |
| 12337.5-c3 |
12337.5-c |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12337.5 |
\( 13^{2} \cdot 73 \) |
\( 13^{6} \cdot 73 \) |
$1.63118$ |
$(4a-3), (-9a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$2.697785033$ |
3.115133830 |
\( \frac{9927}{73} a + \frac{20960}{73} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -5 a + 4\) , \( -5 a + 2\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5a+4\right){x}-5a+2$ |
| 12337.5-c4 |
12337.5-c |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
12337.5 |
\( 13^{2} \cdot 73 \) |
\( 13^{6} \cdot 73^{6} \) |
$1.63118$ |
$(4a-3), (-9a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$0.449630838$ |
3.115133830 |
\( -\frac{55816089234767}{151334226289} a + \frac{107352826006104}{151334226289} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 140 a - 186\) , \( -1021 a - 380\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(140a-186\right){x}-1021a-380$ |
*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.
