| 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 |
| 9.1-a1 |
9.1-a |
$4$ |
$6$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{3} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$1863.874696$ |
1.502124713 |
\( 4189363200 a^{3} + 2075931072 a^{2} - 7387830144 a - 1678779072 \) |
\( \bigl[a^{3} - 4 a - 1\) , \( a^{2} - 2\) , \( a^{2} - 2\) , \( 4 a^{3} - 8 a^{2} + a + 5\) , \( -2 a^{3} + 14 a^{2} - 10 a - 5\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(4a^{3}-8a^{2}+a+5\right){x}-2a^{3}+14a^{2}-10a-5$ |
| 9.1-a2 |
9.1-a |
$4$ |
$6$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$207.0971884$ |
1.502124713 |
\( 2937600 a^{3} - 6516288 a^{2} - 7382016 a + 13364352 \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 2 a - 2\) , \( a^{3} - a^{2} - 2 a + 2\) , \( 9 a^{3} - 8 a^{2} - 36 a - 6\) , \( 60 a^{3} - 47 a^{2} - 238 a - 51\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-2a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-2\right){x}^{2}+\left(9a^{3}-8a^{2}-36a-6\right){x}+60a^{3}-47a^{2}-238a-51$ |
| 9.1-a3 |
9.1-a |
$4$ |
$6$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{3} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$1863.874696$ |
1.502124713 |
\( -4189363200 a^{3} + 14644020672 a^{2} - 9332121600 a - 2801314944 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{2} - a - 2\) , \( a\) , \( -3 a^{3} + a^{2} + 7 a\) , \( a^{3} - 2 a\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-3a^{3}+a^{2}+7a\right){x}+a^{3}-2a$ |
| 9.1-a4 |
9.1-a |
$4$ |
$6$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$207.0971884$ |
1.502124713 |
\( -2937600 a^{3} + 2296512 a^{2} + 11601792 a + 2403648 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 2 a\) , \( a^{3} - a^{2} - 2 a + 1\) , \( -11 a^{3} + 26 a^{2} + 27 a - 52\) , \( -48 a^{3} + 107 a^{2} + 121 a - 220\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-2a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a\right){x}^{2}+\left(-11a^{3}+26a^{2}+27a-52\right){x}-48a^{3}+107a^{2}+121a-220$ |
| 9.1-b1 |
9.1-b |
$6$ |
$18$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1.199731322$ |
$32.34405507$ |
1.125823332 |
\( 44330496 a^{2} - 44330496 a - 11889984 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( 0\) , \( a^{2} - a - 2\) , \( -7 a^{3} + 9 a^{2} + 15 a - 22\) , \( -25 a^{3} + 29 a^{2} + 55 a - 67\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-7a^{3}+9a^{2}+15a-22\right){x}-25a^{3}+29a^{2}+55a-67$ |
| 9.1-b2 |
9.1-b |
$6$ |
$18$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$0.133303480$ |
$2619.868460$ |
1.125823332 |
\( 44330496 a^{2} - 44330496 a - 11889984 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 2 a\) , \( a^{3} - a^{2} - 3 a + 1\) , \( -9 a^{3} + 12 a^{2} + 19 a - 26\) , \( 17 a^{3} - 18 a^{2} - 38 a + 41\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a\right){x}^{2}+\left(-9a^{3}+12a^{2}+19a-26\right){x}+17a^{3}-18a^{2}-38a+41$ |
| 9.1-b3 |
9.1-b |
$6$ |
$18$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \) |
$0.399910440$ |
$873.2894869$ |
1.125823332 |
\( 1728 \) |
\( \bigl[a^{2} - a - 1\) , \( a^{2} - a - 3\) , \( 1\) , \( -a^{2} + a + 3\) , \( -1\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-a^{2}+a+3\right){x}-1$ |
| 9.1-b4 |
9.1-b |
$6$ |
$18$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 2^{2} \) |
$0.199955220$ |
$873.2894869$ |
1.125823332 |
\( 1728 \) |
\( \bigl[a^{2} - a - 1\) , \( a^{2} - a - 3\) , \( a^{2} - a - 2\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}$ |
| 9.1-b5 |
9.1-b |
$6$ |
$18$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2^{2} \) |
$0.599865661$ |
$32.34405507$ |
1.125823332 |
\( -44330496 a^{2} + 44330496 a + 165432000 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( 0\) , \( 1\) , \( -11 a^{3} - a^{2} + 21 a - 4\) , \( -59 a^{3} - 22 a^{2} + 106 a + 10\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+{y}={x}^{3}+\left(-11a^{3}-a^{2}+21a-4\right){x}-59a^{3}-22a^{2}+106a+10$ |
| 9.1-b6 |
9.1-b |
$6$ |
$18$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.066651740$ |
$2619.868460$ |
1.125823332 |
\( -44330496 a^{2} + 44330496 a + 165432000 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 2 a\) , \( a^{3} - 4 a\) , \( -12 a^{3} - a^{2} + 22 a - 4\) , \( 48 a^{3} + 20 a^{2} - 86 a - 14\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a\right){x}^{2}+\left(-12a^{3}-a^{2}+22a-4\right){x}+48a^{3}+20a^{2}-86a-14$ |
| 9.1-c1 |
9.1-c |
$4$ |
$6$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{3} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$183.5898402$ |
1.331620377 |
\( 4189363200 a^{3} + 2075931072 a^{2} - 7387830144 a - 1678779072 \) |
\( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - 5 a - 2\) , \( a^{3} - 3 a\) , \( 3 a^{3} - 9 a^{2} + 4\) , \( 5 a^{3} - 20 a^{2} + 15 a + 2\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(3a^{3}-9a^{2}+4\right){x}+5a^{3}-20a^{2}+15a+2$ |
| 9.1-c2 |
9.1-c |
$4$ |
$6$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$1652.308562$ |
1.331620377 |
\( 2937600 a^{3} - 6516288 a^{2} - 7382016 a + 13364352 \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 2 a - 2\) , \( a^{2} - 1\) , \( 9 a^{3} - 8 a^{2} - 35 a - 6\) , \( -20 a^{3} + 14 a^{2} + 82 a + 17\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-2\right){x}^{2}+\left(9a^{3}-8a^{2}-35a-6\right){x}-20a^{3}+14a^{2}+82a+17$ |
| 9.1-c3 |
9.1-c |
$4$ |
$6$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$1652.308562$ |
1.331620377 |
\( -2937600 a^{3} + 2296512 a^{2} + 11601792 a + 2403648 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 2 a\) , \( a + 1\) , \( -11 a^{3} + 26 a^{2} + 28 a - 51\) , \( 33 a^{3} - 73 a^{2} - 83 a + 149\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a\right){x}^{2}+\left(-11a^{3}+26a^{2}+28a-51\right){x}+33a^{3}-73a^{2}-83a+149$ |
| 9.1-c4 |
9.1-c |
$4$ |
$6$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{3} \) |
$8.10694$ |
$(a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$183.5898402$ |
1.331620377 |
\( -4189363200 a^{3} + 14644020672 a^{2} - 9332121600 a - 2801314944 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} + 3 a + 2\) , \( a\) , \( -4 a^{3} + 11 a + 4\) , \( -4 a^{3} + 10 a + 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+3a+2\right){x}^{2}+\left(-4a^{3}+11a+4\right){x}-4a^{3}+10a+1$ |
*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.
