| 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 |
| 12.1-a1 |
12.1-a |
$2$ |
$2$ |
4.4.13824.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$14.33353$ |
$(a-2), (a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$281.3841477$ |
2.393220776 |
\( -\frac{31739840}{3} a^{3} + 22910944 a^{2} + \frac{40586944}{3} a - \frac{86756704}{3} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + 5 a\) , \( a^{3} + a^{2} - 3 a - 2\) , \( -58 a^{3} + 20 a^{2} + 228 a - 194\) , \( 160 a^{3} - 434 a^{2} - 1023 a + 1475\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(-58a^{3}+20a^{2}+228a-194\right){x}+160a^{3}-434a^{2}-1023a+1475$ |
| 12.1-a2 |
12.1-a |
$2$ |
$2$ |
4.4.13824.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \) |
$14.33353$ |
$(a-2), (a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$140.6920738$ |
2.393220776 |
\( \frac{57088}{9} a^{3} - \frac{76160}{9} a^{2} - \frac{245504}{9} a + \frac{307072}{9} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{3} - a^{2} - 3 a + 4\) , \( a^{2} - 2\) , \( 15 a^{3} + 2 a^{2} - 60 a - 31\) , \( 17 a^{3} - 5 a^{2} - 59 a - 23\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+4\right){x}^{2}+\left(15a^{3}+2a^{2}-60a-31\right){x}+17a^{3}-5a^{2}-59a-23$ |
| 12.1-b1 |
12.1-b |
$2$ |
$2$ |
4.4.13824.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$14.33353$ |
$(a-2), (a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$281.3841477$ |
2.393220776 |
\( \frac{31739840}{3} a^{3} + 22910944 a^{2} - \frac{40586944}{3} a - \frac{86756704}{3} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 5 a\) , \( a^{3} + a^{2} - 3 a - 2\) , \( 57 a^{3} + 20 a^{2} - 228 a - 194\) , \( -161 a^{3} - 434 a^{2} + 1023 a + 1475\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(57a^{3}+20a^{2}-228a-194\right){x}-161a^{3}-434a^{2}+1023a+1475$ |
| 12.1-b2 |
12.1-b |
$2$ |
$2$ |
4.4.13824.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \) |
$14.33353$ |
$(a-2), (a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$140.6920738$ |
2.393220776 |
\( -\frac{57088}{9} a^{3} - \frac{76160}{9} a^{2} + \frac{245504}{9} a + \frac{307072}{9} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} - a^{2} + 3 a + 4\) , \( a^{2} - 2\) , \( -15 a^{3} + 2 a^{2} + 58 a - 31\) , \( -17 a^{3} - 5 a^{2} + 59 a - 23\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+4\right){x}^{2}+\left(-15a^{3}+2a^{2}+58a-31\right){x}-17a^{3}-5a^{2}+59a-23$ |
| 12.1-c1 |
12.1-c |
$2$ |
$2$ |
4.4.13824.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$14.33353$ |
$(a-2), (a^2+a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.019665283$ |
$2234.949414$ |
2.242860625 |
\( \frac{31739840}{3} a^{3} + 22910944 a^{2} - \frac{40586944}{3} a - \frac{86756704}{3} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} - a^{2} + 5 a + 4\) , \( a\) , \( 55 a^{3} + 20 a^{2} - 217 a - 191\) , \( 217 a^{3} + 453 a^{2} - 1246 a - 1668\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}-a^{2}+5a+4\right){x}^{2}+\left(55a^{3}+20a^{2}-217a-191\right){x}+217a^{3}+453a^{2}-1246a-1668$ |
| 12.1-c2 |
12.1-c |
$2$ |
$2$ |
4.4.13824.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \) |
$14.33353$ |
$(a-2), (a^2+a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.039330566$ |
$1117.474707$ |
2.242860625 |
\( -\frac{57088}{9} a^{3} - \frac{76160}{9} a^{2} + \frac{245504}{9} a + \frac{307072}{9} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{2} - 2\) , \( -13 a^{3} + 4 a^{2} + 46 a - 39\) , \( -8 a^{3} + 36 a^{2} + 69 a - 112\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-13a^{3}+4a^{2}+46a-39\right){x}-8a^{3}+36a^{2}+69a-112$ |
| 12.1-d1 |
12.1-d |
$2$ |
$2$ |
4.4.13824.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$14.33353$ |
$(a-2), (a^2+a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.019665283$ |
$2234.949414$ |
2.242860625 |
\( -\frac{31739840}{3} a^{3} + 22910944 a^{2} + \frac{40586944}{3} a - \frac{86756704}{3} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - a^{2} - 5 a + 4\) , \( a\) , \( -56 a^{3} + 20 a^{2} + 219 a - 191\) , \( -217 a^{3} + 453 a^{2} + 1246 a - 1668\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-5a+4\right){x}^{2}+\left(-56a^{3}+20a^{2}+219a-191\right){x}-217a^{3}+453a^{2}+1246a-1668$ |
| 12.1-d2 |
12.1-d |
$2$ |
$2$ |
4.4.13824.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \) |
$14.33353$ |
$(a-2), (a^2+a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.039330566$ |
$1117.474707$ |
2.242860625 |
\( \frac{57088}{9} a^{3} - \frac{76160}{9} a^{2} - \frac{245504}{9} a + \frac{307072}{9} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( a^{2} - 2\) , \( 13 a^{3} + 4 a^{2} - 48 a - 39\) , \( 8 a^{3} + 36 a^{2} - 69 a - 112\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}^{2}+\left(13a^{3}+4a^{2}-48a-39\right){x}+8a^{3}+36a^{2}-69a-112$ |
*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.
