| 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 |
| 10.2-a1 |
10.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{7} \cdot 5 \) |
$1.05406$ |
$(a+3), (-a-4)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$43.11092390$ |
0.722135146 |
\( \frac{378329}{80} a + \frac{707609}{80} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 8 a - 32\) , \( -15 a + 47\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(8a-32\right){x}-15a+47$ |
| 10.2-a2 |
10.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{21} \cdot 5^{3} \) |
$1.05406$ |
$(a+3), (-a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$4.790102655$ |
0.722135146 |
\( \frac{4874397503}{256000} a + \frac{16590792793}{256000} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -72 a + 233\) , \( -47 a + 153\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-72a+233\right){x}-47a+153$ |
| 10.2-b1 |
10.2-b |
$2$ |
$7$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{7} \cdot 5^{7} \) |
$1.05406$ |
$(a+3), (-a-4)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 7^{2} \) |
$1$ |
$11.81659665$ |
1.781418972 |
\( \frac{584688139}{1250000} a + \frac{1940357659}{1250000} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -15 a + 49\) , \( -2176 a + 7212\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-15a+49\right){x}-2176a+7212$ |
| 10.2-b2 |
10.2-b |
$2$ |
$7$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5 \) |
$1.05406$ |
$(a+3), (-a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$49$ |
\( 1 \) |
$1$ |
$0.241155033$ |
1.781418972 |
\( \frac{466209435421917067326607}{10} a + \frac{1546241771017925012924657}{10} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 23540 a - 78106\) , \( 3623859 a - 12019033\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(23540a-78106\right){x}+3623859a-12019033$ |
| 10.2-c1 |
10.2-c |
$2$ |
$7$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{7} \cdot 5^{7} \) |
$1.05406$ |
$(a+3), (-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.6.1 |
$1$ |
\( 7 \) |
$0.058791529$ |
$8.658898230$ |
1.074432388 |
\( \frac{584688139}{1250000} a + \frac{1940357659}{1250000} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -13 a + 50\) , \( 2162 a - 7166\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-13a+50\right){x}+2162a-7166$ |
| 10.2-c2 |
10.2-c |
$2$ |
$7$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5 \) |
$1.05406$ |
$(a+3), (-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.6.3 |
$1$ |
\( 1 \) |
$0.411540706$ |
$8.658898230$ |
1.074432388 |
\( \frac{466209435421917067326607}{10} a + \frac{1546241771017925012924657}{10} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 23542 a - 78105\) , \( -3600318 a + 11940924\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(23542a-78105\right){x}-3600318a+11940924$ |
| 10.2-d1 |
10.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{7} \cdot 5 \) |
$1.05406$ |
$(a+3), (-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 7 \) |
$0.082400613$ |
$9.456690263$ |
1.644641729 |
\( \frac{378329}{80} a + \frac{707609}{80} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 10 a - 31\) , \( 24 a - 79\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(10a-31\right){x}+24a-79$ |
| 10.2-d2 |
10.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{21} \cdot 5^{3} \) |
$1.05406$ |
$(a+3), (-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \cdot 7 \) |
$0.027466871$ |
$9.456690263$ |
1.644641729 |
\( \frac{4874397503}{256000} a + \frac{16590792793}{256000} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -70 a + 234\) , \( -24 a + 80\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-70a+234\right){x}-24a+80$ |
*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.
