| 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 |
| 18688.1-a1 |
18688.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{20} \cdot 73^{2} \) |
$1.80963$ |
$(-9a+1), (2)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.137241188$ |
$1.901095323$ |
2.410170229 |
\( \frac{55351840}{5329} a - \frac{65747556}{5329} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 18 a - 21\) , \( -47 a + 29\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(18a-21\right){x}-47a+29$ |
| 18688.1-a2 |
18688.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{16} \cdot 73 \) |
$1.80963$ |
$(-9a+1), (2)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.137241188$ |
$3.802190646$ |
2.410170229 |
\( \frac{10496}{73} a - \frac{51248}{73} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -2 a - 1\) , \( -3 a + 1\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a-1\right){x}-3a+1$ |
| 18688.1-b1 |
18688.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{8} \cdot 73 \) |
$1.80963$ |
$(-9a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.694406370$ |
$5.343798536$ |
2.142412757 |
\( -\frac{1048576}{73} a - \frac{131072}{73} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 3\) , \( 2 a\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+3{x}+2a$ |
| 18688.1-b2 |
18688.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{16} \cdot 73^{2} \) |
$1.80963$ |
$(-9a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.347203185$ |
$2.671899268$ |
2.142412757 |
\( -\frac{1251200}{5329} a - \frac{1700496}{5329} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -5 a + 3\) , \( 5 a - 7\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-5a+3\right){x}+5a-7$ |
| 18688.1-c1 |
18688.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{22} \cdot 73 \) |
$1.80963$ |
$(-9a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.526722904$ |
$0.928235668$ |
2.708228088 |
\( \frac{19225018386}{73} a - \frac{24487503120}{73} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -340 a + 384\) , \( -572 a - 2168\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-340a+384\right){x}-572a-2168$ |
| 18688.1-c2 |
18688.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{16} \cdot 73 \) |
$1.80963$ |
$(-9a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.631680726$ |
$3.712942672$ |
2.708228088 |
\( \frac{190512}{73} a - \frac{43200}{73} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 4\) , \( 4 a\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+4{x}+4a$ |
| 18688.1-c3 |
18688.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{22} \cdot 73^{4} \) |
$1.80963$ |
$(-9a+1), (2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.526722904$ |
$0.928235668$ |
2.708228088 |
\( -\frac{3793736898}{28398241} a - \frac{7074562464}{28398241} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -20 a - 16\) , \( -156 a - 56\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-20a-16\right){x}-156a-56$ |
| 18688.1-c4 |
18688.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{20} \cdot 73^{2} \) |
$1.80963$ |
$(-9a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.263361452$ |
$1.856471336$ |
2.708228088 |
\( -\frac{98161308}{5329} a + \frac{14950332}{5329} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -20 a + 24\) , \( -12 a - 24\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-20a+24\right){x}-12a-24$ |
| 18688.1-d1 |
18688.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{8} \cdot 73 \) |
$1.80963$ |
$(-9a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1.174732303$ |
$5.343798536$ |
3.624335231 |
\( -\frac{1048576}{73} a - \frac{131072}{73} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 3 a - 3\) , \( -2 a\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(3a-3\right){x}-2a$ |
| 18688.1-d2 |
18688.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{16} \cdot 73^{2} \) |
$1.80963$ |
$(-9a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.587366151$ |
$2.671899268$ |
3.624335231 |
\( -\frac{1251200}{5329} a - \frac{1700496}{5329} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 3 a + 2\) , \( -5 a + 7\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(3a+2\right){x}-5a+7$ |
| 18688.1-e1 |
18688.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{24} \cdot 73^{3} \) |
$1.80963$ |
$(-9a+1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$0.810583522$ |
2.807943688 |
\( \frac{60988685561}{389017} a - \frac{169775626841}{389017} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -234 a + 77\) , \( 1113 a - 1050\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-234a+77\right){x}+1113a-1050$ |
| 18688.1-e2 |
18688.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{24} \cdot 73^{2} \) |
$1.80963$ |
$(-9a+1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$1.215875283$ |
2.807943688 |
\( -\frac{927841113}{5329} a - \frac{395933743}{5329} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 86 a - 83\) , \( 345 a - 186\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(86a-83\right){x}+345a-186$ |
| 18688.1-e3 |
18688.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{24} \cdot 73 \) |
$1.80963$ |
$(-9a+1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$2.431750566$ |
2.807943688 |
\( \frac{9927}{73} a + \frac{20960}{73} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 6 a - 3\) , \( 9 a - 10\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(6a-3\right){x}+9a-10$ |
| 18688.1-e4 |
18688.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
18688.1 |
\( 2^{8} \cdot 73 \) |
\( 2^{24} \cdot 73^{6} \) |
$1.80963$ |
$(-9a+1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$0.405291761$ |
2.807943688 |
\( -\frac{55816089234767}{151334226289} a + \frac{107352826006104}{151334226289} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -234 a + 157\) , \( 1849 a - 714\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-234a+157\right){x}+1849a-714$ |
*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.
