| 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 |
| 21.1-a2 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{48} \cdot 7^{6} \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$16$ |
\( 2^{2} \) |
$1$ |
$2.052706454$ |
0.742422999 |
\( -\frac{91305470986913768828360282696367059}{9384442261550973912914558289} a^{3} - \frac{36191380838537641287089198334152081}{9384442261550973912914558289} a^{2} + \frac{116960169615559391050979984902302073}{3128147420516991304304852763} a + \frac{230385512085604356994786103257239902}{9384442261550973912914558289} \) |
\( \bigl[a^{2} - 1\) , \( a^{3} - a^{2} - 2 a + 1\) , \( a^{3} - 3 a - 1\) , \( 312 a^{3} - 201 a^{2} - 1090 a + 375\) , \( -2482 a^{3} + 1995 a^{2} + 8193 a - 3631\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+1\right){x}^{2}+\left(312a^{3}-201a^{2}-1090a+375\right){x}-2482a^{3}+1995a^{2}+8193a-3631$ |
| 63.1-b5 |
63.1-b |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
63.1 |
\( 3^{2} \cdot 7 \) |
\( - 3^{54} \cdot 7^{6} \) |
$6.63516$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$2.116500526$ |
1.148244065 |
\( -\frac{91305470986913768828360282696367059}{9384442261550973912914558289} a^{3} - \frac{36191380838537641287089198334152081}{9384442261550973912914558289} a^{2} + \frac{116960169615559391050979984902302073}{3128147420516991304304852763} a + \frac{230385512085604356994786103257239902}{9384442261550973912914558289} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 3 a\) , \( a^{3} - 3 a - 1\) , \( 69 a^{3} + 32 a^{2} - 124 a - 55\) , \( -113 a^{3} - 3242 a^{2} - 698 a + 1061\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(69a^{3}+32a^{2}-124a-55\right){x}-113a^{3}-3242a^{2}-698a+1061$ |
| 147.1-b11 |
147.1-b |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( - 3^{48} \cdot 7^{12} \) |
$7.37647$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{5} \cdot 3 \) |
$1$ |
$0.747857152$ |
1.622910133 |
\( -\frac{91305470986913768828360282696367059}{9384442261550973912914558289} a^{3} - \frac{36191380838537641287089198334152081}{9384442261550973912914558289} a^{2} + \frac{116960169615559391050979984902302073}{3128147420516991304304852763} a + \frac{230385512085604356994786103257239902}{9384442261550973912914558289} \) |
\( \bigl[a + 1\) , \( -a^{3} + 3 a + 1\) , \( a^{3} - 3 a\) , \( 108 a^{3} + 68 a^{2} - 170 a - 18\) , \( 3266 a^{3} + 3897 a^{2} - 521 a - 901\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(108a^{3}+68a^{2}-170a-18\right){x}+3266a^{3}+3897a^{2}-521a-901$ |
| 441.1-c3 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{54} \cdot 7^{12} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$16$ |
\( 2^{3} \) |
$1$ |
$1.866360265$ |
1.350050597 |
\( -\frac{91305470986913768828360282696367059}{9384442261550973912914558289} a^{3} - \frac{36191380838537641287089198334152081}{9384442261550973912914558289} a^{2} + \frac{116960169615559391050979984902302073}{3128147420516991304304852763} a + \frac{230385512085604356994786103257239902}{9384442261550973912914558289} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a\) , \( a^{2} - a - 2\) , \( 653 a^{3} - 161 a^{2} - 2351 a - 135\) , \( -7041 a^{3} + 1756 a^{2} + 24858 a + 3620\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(653a^{3}-161a^{2}-2351a-135\right){x}-7041a^{3}+1756a^{2}+24858a+3620$ |
*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.
