| 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 |
| 3.1-a1 |
3.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
3.1 |
\( 3 \) |
\( 3^{6} \cdot 7^{12} \) |
$1.23903$ |
$(3,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Cn |
$1$ |
\( 2 \) |
$3.388142012$ |
$8.084873186$ |
1.299999940 |
\( \frac{1331}{27} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( 2 a - 5\) , \( 15 a + 11\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-a{x}^2+\left(2a-5\right){x}+15a+11$ |
| 3.1-b1 |
3.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
3.1 |
\( 3 \) |
\( 3^{6} \cdot 7^{12} \) |
$1.23903$ |
$(3,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Cn |
$1$ |
\( 2 \) |
$3.388142012$ |
$8.084873186$ |
1.299999940 |
\( \frac{1331}{27} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -2 a - 3\) , \( -15 a + 26\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(a-1\right){x}^2+\left(-2a-3\right){x}-15a+26$ |
| 144.1-b1 |
144.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
144.1 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{24} \cdot 3^{12} \) |
$3.26130$ |
$(2,a), (3,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cn |
$1$ |
\( 2^{4} \) |
$2.095922209$ |
$4.667803710$ |
1.857189633 |
\( \frac{1331}{27} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a - 20\) , \( 20 a\bigr] \) |
${y}^2={x}^3+\left(-a-1\right){x}^2+\left(a-20\right){x}+20a$ |
| 144.1-c1 |
144.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
144.1 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{12} \cdot 5^{12} \) |
$3.26130$ |
$(2,a), (3,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cn |
$1$ |
\( 2^{4} \) |
$8.534581015$ |
$4.667803710$ |
7.562463586 |
\( \frac{1331}{27} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -17 a - 23\) , \( -7 a - 1050\bigr] \) |
${y}^2+a{x}{y}={x}^3+a{x}^2+\left(-17a-23\right){x}-7a-1050$ |
| 144.5-b1 |
144.5-b |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
144.5 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{24} \cdot 3^{12} \) |
$3.26130$ |
$(2,a+1), (3,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cn |
$1$ |
\( 2^{4} \) |
$2.095922209$ |
$4.667803710$ |
1.857189633 |
\( \frac{1331}{27} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a - 20\) , \( -20 a\bigr] \) |
${y}^2={x}^3+\left(a+1\right){x}^2+\left(a-20\right){x}-20a$ |
| 144.5-c1 |
144.5-c |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
144.5 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{12} \cdot 5^{12} \) |
$3.26130$ |
$(2,a+1), (3,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cn |
$1$ |
\( 2^{4} \) |
$8.534581015$ |
$4.667803710$ |
7.562463586 |
\( \frac{1331}{27} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 6 a - 54\) , \( -15 a - 1365\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a+1\right){x}^2+\left(6a-54\right){x}-15a-1365$ |
| 441.1-b1 |
441.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{12} \cdot 5^{12} \cdot 7^{6} \) |
$4.31429$ |
$(3,a+1), (7,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cn |
$4$ |
\( 2^{4} \) |
$0.798967479$ |
$3.528527939$ |
2.140677088 |
\( \frac{1331}{27} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -22 a + 34\) , \( -519 a - 617\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+a{x}^2+\left(-22a+34\right){x}-519a-617$ |
| 441.1-c1 |
441.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 2^{12} \cdot 3^{12} \cdot 7^{6} \) |
$4.31429$ |
$(3,a+1), (7,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cn |
$4$ |
\( 2^{4} \) |
$2.345068151$ |
$3.528527939$ |
6.283151429 |
\( \frac{1331}{27} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -2 a + 40\) , \( -5 a + 120\bigr] \) |
${y}^2+a{x}{y}={x}^3-{x}^2+\left(-2a+40\right){x}-5a+120$ |
| 441.3-b1 |
441.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
441.3 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{12} \cdot 5^{12} \cdot 7^{6} \) |
$4.31429$ |
$(3,a+1), (7,a+6)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cn |
$4$ |
\( 2^{4} \) |
$0.798967479$ |
$3.528527939$ |
2.140677088 |
\( \frac{1331}{27} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 21 a + 12\) , \( 518 a - 1136\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(21a+12\right){x}+518a-1136$ |
| 441.3-c1 |
441.3-c |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
441.3 |
\( 3^{2} \cdot 7^{2} \) |
\( 2^{12} \cdot 3^{12} \cdot 7^{6} \) |
$4.31429$ |
$(3,a+1), (7,a+6)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cn |
$4$ |
\( 2^{4} \) |
$2.345068151$ |
$3.528527939$ |
6.283151429 |
\( \frac{1331}{27} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 2 a + 38\) , \( 5 a + 115\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a-1\right){x}^2+\left(2a+38\right){x}+5a+115$ |
| 768.5-r1 |
768.5-r |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
768.5 |
\( 2^{8} \cdot 3 \) |
\( 2^{24} \cdot 3^{6} \cdot 7^{12} \) |
$4.95610$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3Cn |
$1$ |
\( 2^{5} \cdot 3 \) |
$1$ |
$4.042436593$ |
2.302146608 |
\( \frac{1331}{27} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -29 a + 97\) , \( 1002 a - 882\bigr] \) |
${y}^2={x}^3-a{x}^2+\left(-29a+97\right){x}+1002a-882$ |
| 768.5-s1 |
768.5-s |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
768.5 |
\( 2^{8} \cdot 3 \) |
\( 2^{24} \cdot 3^{6} \cdot 7^{12} \) |
$4.95610$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3Cn |
$1$ |
\( 2^{5} \cdot 3 \) |
$1$ |
$4.042436593$ |
2.302146608 |
\( \frac{1331}{27} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 29 a + 68\) , \( -1002 a + 120\bigr] \) |
${y}^2={x}^3+\left(a-1\right){x}^2+\left(29a+68\right){x}-1002a+120$ |
*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.
