| 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 |
| 8.1-a1 |
8.1-a |
$2$ |
$2$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{6} \) |
$12.33175$ |
$(a^3-5a), (-a^2+a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$340.3879291$ |
3.198705229 |
\( \frac{1404792715}{4} a^{3} - \frac{1905362155}{4} a^{2} - 1586175900 a + 1970002986 \) |
\( \bigl[a^{3} + a^{2} - 4 a - 3\) , \( a^{2} - 2\) , \( a^{2} - 2\) , \( -6 a^{3} + 7 a^{2} + 40 a - 41\) , \( -12 a^{3} + 10 a^{2} + 90 a - 97\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(-6a^{3}+7a^{2}+40a-41\right){x}-12a^{3}+10a^{2}+90a-97$ |
| 8.1-a2 |
8.1-a |
$2$ |
$2$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{6} \) |
$12.33175$ |
$(a^3-5a), (-a^2+a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$340.3879291$ |
3.198705229 |
\( -\frac{28279490875}{4} a^{3} - \frac{2685693575}{2} a^{2} + \frac{67502881775}{2} a + \frac{47530715137}{4} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} + a^{2} - 5 a - 4\) , \( a^{3} + a^{2} - 3 a - 2\) , \( -2 a - 2\) , \( -4 a^{3} - 20 a^{2} + 50 a + 20\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-4\right){x}^{2}+\left(-2a-2\right){x}-4a^{3}-20a^{2}+50a+20$ |
| 8.1-b1 |
8.1-b |
$1$ |
$1$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{9} \) |
$12.33175$ |
$(a^3-5a), (-a^2+a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$95.00683401$ |
0.892801509 |
\( -\frac{1448503}{16} a^{3} + \frac{2031095}{16} a^{2} + \frac{6530981}{16} a - \frac{4219149}{8} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 2\) , \( a^{2} - 3\) , \( a^{3} - 5 a - 1\) , \( a^{3} - 5 a - 2\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(a^{3}-5a-1\right){x}+a^{3}-5a-2$ |
| 8.1-c1 |
8.1-c |
$4$ |
$4$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$12.33175$ |
$(a^3-5a), (-a^2+a+3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.093613475$ |
$1064.136859$ |
1.872258966 |
\( \frac{1148659}{16} a^{3} - 97391 a^{2} - \frac{2597693}{8} a + \frac{1618985}{4} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 3\) , \( -a^{3} - a^{2} + 3 a + 3\) , \( a\) , \( -7 a^{3} + 32 a + 4\) , \( -5 a^{3} - 2 a^{2} + 24 a + 12\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}-a^{2}+3a+3\right){x}^{2}+\left(-7a^{3}+32a+4\right){x}-5a^{3}-2a^{2}+24a+12$ |
| 8.1-c2 |
8.1-c |
$4$ |
$4$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$12.33175$ |
$(a^3-5a), (-a^2+a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.374453900$ |
$266.0342149$ |
1.872258966 |
\( -802630664191 a^{3} + 2479833481214 a^{2} - 1168780558340 a - \frac{1536406845933}{2} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 3\) , \( a^{2} + a - 3\) , \( a^{3} - 4 a\) , \( 242 a^{3} - 718 a^{2} + 323 a + 220\) , \( 6877 a^{3} - 21198 a^{2} + 9972 a + 6560\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-3\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(242a^{3}-718a^{2}+323a+220\right){x}+6877a^{3}-21198a^{2}+9972a+6560$ |
| 8.1-c3 |
8.1-c |
$4$ |
$4$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$12.33175$ |
$(a^3-5a), (-a^2+a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.374453900$ |
$266.0342149$ |
1.872258966 |
\( -\frac{174483}{16} a^{3} - 234 a^{2} + \frac{466917}{8} a + \frac{352373}{16} \) |
\( \bigl[a^{3} - 3 a + 1\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{3} - 4 a + 1\) , \( -a^{3} + a^{2} - 1\) , \( -3 a^{3} - 2 a^{2} + 5 a\bigr] \) |
${y}^2+\left(a^{3}-3a+1\right){x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(-a^{3}+a^{2}-1\right){x}-3a^{3}-2a^{2}+5a$ |
| 8.1-c4 |
8.1-c |
$4$ |
$4$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$12.33175$ |
$(a^3-5a), (-a^2+a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.187226950$ |
$1064.136859$ |
1.872258966 |
\( -180162 a^{3} + 556938 a^{2} - 262398 a - \frac{682855}{4} \) |
\( \bigl[1\) , \( 0\) , \( a^{3} + a^{2} - 3 a - 3\) , \( -3 a^{3} + 4 a^{2} + 12 a - 19\) , \( -2 a^{2} - 2 a + 4\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}+a^{2}-3a-3\right){y}={x}^{3}+\left(-3a^{3}+4a^{2}+12a-19\right){x}-2a^{2}-2a+4$ |
| 8.1-d1 |
8.1-d |
$3$ |
$9$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{27} \) |
$12.33175$ |
$(a^3-5a), (-a^2+a+3)$ |
0 |
$\Z/9\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{4} \) |
$1$ |
$180.3949705$ |
1.695213861 |
\( -\frac{3175}{256} a^{3} + \frac{162185}{512} a^{2} - \frac{16573}{256} a - \frac{87921}{256} \) |
\( \bigl[1\) , \( a^{2} + a - 3\) , \( a^{3} - 4 a + 1\) , \( a^{3} + 3 a^{2} - 4 a\) , \( 2 a^{2} - a - 3\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(a^{3}+3a^{2}-4a\right){x}+2a^{2}-a-3$ |
| 8.1-d2 |
8.1-d |
$3$ |
$9$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{57} \) |
$12.33175$ |
$(a^3-5a), (-a^2+a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$9$ |
\( 3^{4} \) |
$1$ |
$2.227098402$ |
1.695213861 |
\( \frac{112807378703683}{134217728} a^{3} + \frac{10131794073095}{67108864} a^{2} - \frac{270571608370495}{67108864} a - \frac{11896758820741}{8388608} \) |
\( \bigl[1\) , \( a^{2} + a - 3\) , \( a^{3} - 4 a + 1\) , \( -14 a^{3} - 47 a^{2} + 36 a + 20\) , \( -169 a^{3} - 350 a^{2} + 445 a + 193\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(-14a^{3}-47a^{2}+36a+20\right){x}-169a^{3}-350a^{2}+445a+193$ |
| 8.1-d3 |
8.1-d |
$3$ |
$9$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{19} \) |
$12.33175$ |
$(a^3-5a), (-a^2+a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$729$ |
\( 3^{2} \) |
$1$ |
$0.027495042$ |
1.695213861 |
\( \frac{9135977966106098429399}{512} a^{3} + \frac{867628940947068681587}{256} a^{2} - \frac{21807541432668180905747}{256} a - \frac{959710996587311585475}{32} \) |
\( \bigl[1\) , \( a^{2} + a - 3\) , \( a^{3} - 4 a + 1\) , \( -1149 a^{3} - 3922 a^{2} + 3071 a + 1540\) , \( -99521 a^{3} - 216742 a^{2} + 264677 a + 117393\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(-1149a^{3}-3922a^{2}+3071a+1540\right){x}-99521a^{3}-216742a^{2}+264677a+117393$ |
*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.
