| 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 |
| 20164.1-a1 |
20164.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20164.1 |
\( 2^{2} \cdot 71^{2} \) |
\( 2^{18} \cdot 71^{2} \) |
$1.84435$ |
$(2), (71)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 3^{2} \) |
$0.033789142$ |
$2.268679151$ |
1.593280077 |
\( \frac{176558481}{36352} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -12 a + 11\) , \( 15\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-12a+11\right){x}+15$ |
| 20164.1-b1 |
20164.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20164.1 |
\( 2^{2} \cdot 71^{2} \) |
\( 2^{2} \cdot 71^{2} \) |
$1.84435$ |
$(2), (71)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.180913710$ |
$5.890151194$ |
2.460918827 |
\( \frac{389017}{142} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( -a\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}-a{x}$ |
| 20164.1-c1 |
20164.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20164.1 |
\( 2^{2} \cdot 71^{2} \) |
\( 2^{12} \cdot 71^{2} \) |
$1.84435$ |
$(2), (71)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.266412564$ |
$3.438653079$ |
3.173464819 |
\( -\frac{185193}{4544} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -a + 1\) , \( -3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}-3$ |
| 20164.1-c2 |
20164.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20164.1 |
\( 2^{2} \cdot 71^{2} \) |
\( 2^{6} \cdot 71^{4} \) |
$1.84435$ |
$(2), (71)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.532825128$ |
$1.719326539$ |
3.173464819 |
\( \frac{7727161833}{40328} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -41 a + 41\) , \( -91\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-41a+41\right){x}-91$ |
| 20164.1-d1 |
20164.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20164.1 |
\( 2^{2} \cdot 71^{2} \) |
\( 2^{2} \cdot 71^{6} \) |
$1.84435$ |
$(2), (71)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1[2] |
$1$ |
\( 3 \) |
$3.625113298$ |
$1.230074669$ |
3.432663002 |
\( \frac{21601086625}{715822} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 58 a\) , \( -170\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+58a{x}-170$ |
| 20164.1-d2 |
20164.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20164.1 |
\( 2^{2} \cdot 71^{2} \) |
\( 2^{6} \cdot 71^{2} \) |
$1.84435$ |
$(2), (71)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1[2] |
$1$ |
\( 3 \) |
$1.208371099$ |
$3.690224009$ |
3.432663002 |
\( \frac{57066625}{568} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 8 a\) , \( 8\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+8a{x}+8$ |
| 20164.1-e1 |
20164.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20164.1 |
\( 2^{2} \cdot 71^{2} \) |
\( 2^{54} \cdot 71^{2} \) |
$1.84435$ |
$(2), (71)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 3^{3} \) |
$0.317904479$ |
$0.216950885$ |
4.300522286 |
\( \frac{2003092024307193}{9529458688} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 2626 a\) , \( 52244\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+2626a{x}+52244$ |
*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.
