| 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 |
| 45.1-a1 |
45.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 2^{12} \cdot 3^{32} \cdot 5^{2} \) |
$1.86594$ |
$(5,a+2), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$16.45312041$ |
$0.490422220$ |
4.003333110 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -986 a - 3512\) , \( -67741 a - 239296\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-986a-3512\right){x}-67741a-239296$ |
| 45.1-a2 |
45.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 2^{12} \cdot 3^{2} \cdot 5^{2} \) |
$1.86594$ |
$(5,a+2), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.028320025$ |
$31.38702211$ |
4.003333110 |
\( -\frac{1}{15} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 4 a + 8\) , \( 19 a + 64\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a+8\right){x}+19a+64$ |
| 45.1-a3 |
45.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 2^{12} \cdot 3^{4} \cdot 5^{16} \) |
$1.86594$ |
$(5,a+2), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$2.056640051$ |
$1.961688882$ |
4.003333110 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 319 a + 1128\) , \( -2137 a - 7552\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(319a+1128\right){x}-2137a-7552$ |
| 45.1-a4 |
45.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 2^{12} \cdot 3^{8} \cdot 5^{8} \) |
$1.86594$ |
$(5,a+2), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$4.113280102$ |
$7.846755528$ |
4.003333110 |
\( \frac{111284641}{50625} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -86 a - 312\) , \( -751 a - 2656\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-86a-312\right){x}-751a-2656$ |
| 45.1-a5 |
45.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 2^{12} \cdot 3^{4} \cdot 5^{4} \) |
$1.86594$ |
$(5,a+2), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$2.056640051$ |
$31.38702211$ |
4.003333110 |
\( \frac{13997521}{225} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -41 a - 152\) , \( 173 a + 608\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-41a-152\right){x}+173a+608$ |
| 45.1-a6 |
45.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 2^{12} \cdot 3^{16} \cdot 5^{4} \) |
$1.86594$ |
$(5,a+2), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$8.226560205$ |
$1.961688882$ |
4.003333110 |
\( \frac{272223782641}{164025} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -1211 a - 4312\) , \( -50801 a - 179456\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1211a-4312\right){x}-50801a-179456$ |
| 45.1-a7 |
45.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 2^{12} \cdot 3^{2} \cdot 5^{2} \) |
$1.86594$ |
$(5,a+2), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.028320025$ |
$31.38702211$ |
4.003333110 |
\( \frac{56667352321}{15} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -716 a - 2552\) , \( 18653 a + 65888\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-716a-2552\right){x}+18653a+65888$ |
| 45.1-a8 |
45.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 2^{12} \cdot 3^{8} \cdot 5^{2} \) |
$1.86594$ |
$(5,a+2), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$4.113280102$ |
$0.490422220$ |
4.003333110 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -19436 a - 69112\) , \( -3044561 a - 10754816\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-19436a-69112\right){x}-3044561a-10754816$ |
| 45.1-b1 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{32} \cdot 5^{2} \) |
$1.86594$ |
$(5,a+2), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.490422220$ |
0.486635119 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -110\) , \( -880\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-110{x}-880$ |
| 45.1-b2 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$1.86594$ |
$(5,a+2), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$31.38702211$ |
0.486635119 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}$ |
| 45.1-b3 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{16} \) |
$1.86594$ |
$(5,a+2), (3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{5} \) |
$1$ |
$1.961688882$ |
0.486635119 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 35\) , \( -28\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+35{x}-28$ |
| 45.1-b4 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$1.86594$ |
$(5,a+2), (3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$7.846755528$ |
0.486635119 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -10\) , \( -10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-10{x}-10$ |
| 45.1-b5 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$1.86594$ |
$(5,a+2), (3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$31.38702211$ |
0.486635119 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -5\) , \( 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-5{x}+2$ |
| 45.1-b6 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$1.86594$ |
$(5,a+2), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.961688882$ |
0.486635119 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -135\) , \( -660\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-135{x}-660$ |
| 45.1-b7 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$1.86594$ |
$(5,a+2), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$31.38702211$ |
0.486635119 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-80{x}+242$ |
| 45.1-b8 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{65}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$1.86594$ |
$(5,a+2), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$0.490422220$ |
0.486635119 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -2160\) , \( -39540\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-2160{x}-39540$ |
*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.
