| 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 |
| 1728.2-a3 |
1728.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
1728.2 |
\( 2^{6} \cdot 3^{3} \) |
\( 2^{6} \cdot 3^{31} \) |
$1.62956$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$2.798543047$ |
$0.561726762$ |
2.223167090 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 7 a - 128\) , \( 73 a + 816\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(7a-128\right){x}+73a+816$ |
| 1728.2-b3 |
1728.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
1728.2 |
\( 2^{6} \cdot 3^{3} \) |
\( 2^{6} \cdot 3^{31} \) |
$1.62956$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.561726762$ |
1.986004013 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 7 a - 128\) , \( -73 a - 816\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(7a-128\right){x}-73a-816$ |
| 1728.3-a3 |
1728.3-a |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
1728.3 |
\( 2^{6} \cdot 3^{3} \) |
\( 2^{6} \cdot 3^{31} \) |
$1.62956$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$2.798543047$ |
$0.561726762$ |
2.223167090 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -62 a + 93\) , \( -416 a - 651\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-62a+93\right){x}-416a-651$ |
| 1728.3-b3 |
1728.3-b |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
1728.3 |
\( 2^{6} \cdot 3^{3} \) |
\( 2^{6} \cdot 3^{31} \) |
$1.62956$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.561726762$ |
1.986004013 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -62 a + 93\) , \( 416 a + 652\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-62a+93\right){x}+416a+652$ |
| 2304.2-d3 |
2304.2-d |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
2304.2 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{18} \cdot 3^{25} \) |
$1.75107$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.486469646$ |
1.719929928 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 110 a + 69\) , \( 433 a + 1042\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(110a+69\right){x}+433a+1042$ |
| 2304.2-f3 |
2304.2-f |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
2304.2 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{18} \cdot 3^{25} \) |
$1.75107$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.486469646$ |
1.719929928 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 110 a + 69\) , \( -433 a - 1042\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(110a+69\right){x}-433a-1042$ |
| 9216.2-b3 |
9216.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
9216.2 |
\( 2^{10} \cdot 3^{2} \) |
\( 2^{24} \cdot 3^{25} \) |
$2.47639$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$5.666583769$ |
$0.343985985$ |
2.756621001 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -220 a - 139\) , \( 2304 a - 1593\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-220a-139\right){x}+2304a-1593$ |
| 9216.2-z3 |
9216.2-z |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
9216.2 |
\( 2^{10} \cdot 3^{2} \) |
\( 2^{24} \cdot 3^{25} \) |
$2.47639$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{4} \cdot 5^{2} \) |
$0.112241099$ |
$0.343985985$ |
5.460188802 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -220 a - 139\) , \( -2304 a + 1593\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-220a-139\right){x}-2304a+1593$ |
| 20736.3-k3 |
20736.3-k |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
20736.3 |
\( 2^{8} \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{37} \) |
$3.03295$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{5} \) |
$2.500738767$ |
$0.162156548$ |
4.587835145 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 990 a + 627\) , \( 11066 a + 30114\bigr] \) |
${y}^2={x}^{3}+\left(990a+627\right){x}+11066a+30114$ |
| 20736.3-l3 |
20736.3-l |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
20736.3 |
\( 2^{8} \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{37} \) |
$3.03295$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{5} \) |
$2.500738767$ |
$0.162156548$ |
4.587835145 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 990 a + 627\) , \( -11066 a - 30114\bigr] \) |
${y}^2={x}^{3}+\left(990a+627\right){x}-11066a-30114$ |
| 27648.2-o3 |
27648.2-o |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.2 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{24} \cdot 3^{31} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{5} \) |
$1.865702059$ |
$0.198600401$ |
4.192059108 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -58 a + 1019\) , \( 13113 a - 3357\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-58a+1019\right){x}+13113a-3357$ |
| 27648.2-bi3 |
27648.2-bi |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.2 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{24} \cdot 3^{31} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.198600401$ |
2.808633811 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -58 a + 1019\) , \( -13113 a + 3357\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-58a+1019\right){x}-13113a+3357$ |
| 27648.3-n3 |
27648.3-n |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.3 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{24} \cdot 3^{31} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{3} \) |
$7.462808236$ |
$0.198600401$ |
4.192059108 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 498 a - 741\) , \( 9927 a - 12573\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(498a-741\right){x}+9927a-12573$ |
| 27648.3-bh3 |
27648.3-bh |
$8$ |
$20$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.3 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{24} \cdot 3^{31} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.198600401$ |
2.808633811 |
\( \frac{2514081593672}{3486784401} a - \frac{1943385699640}{3486784401} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 498 a - 741\) , \( -9927 a + 12573\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(498a-741\right){x}-9927a+12573$ |
*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.
