| 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 |
| 40.1-a1 |
40.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{20} \cdot 5^{8} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$4.406960782$ |
1.137872381 |
\( \frac{237276}{625} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 13\) , \( -34\bigr] \) |
${y}^2={x}^{3}+13{x}-34$ |
| 40.1-a2 |
40.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{16} \cdot 5^{4} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$17.62784313$ |
1.137872381 |
\( \frac{148176}{25} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -7\) , \( -6\bigr] \) |
${y}^2={x}^{3}-7{x}-6$ |
| 40.1-a3 |
40.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$35.25568626$ |
1.137872381 |
\( \frac{55296}{5} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2\) , \( 1\bigr] \) |
${y}^2={x}^{3}-2{x}+1$ |
| 40.1-a4 |
40.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{20} \cdot 5^{2} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$4.406960782$ |
1.137872381 |
\( \frac{132304644}{5} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -107\) , \( -426\bigr] \) |
${y}^2={x}^{3}-107{x}-426$ |
| 40.1-b1 |
40.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{8} \cdot 5^{8} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.638218348$ |
$8.151961419$ |
2.686678917 |
\( \frac{237276}{625} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -23 a + 115\) , \( -240 a + 955\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-23a+115\right){x}-240a+955$ |
| 40.1-b2 |
40.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{4} \cdot 5^{4} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.638218348$ |
$32.60784567$ |
2.686678917 |
\( \frac{148176}{25} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 17 a - 40\) , \( -57 a + 246\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(17a-40\right){x}-57a+246$ |
| 40.1-b3 |
40.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{20} \cdot 5^{2} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.159554587$ |
$16.30392283$ |
2.686678917 |
\( \frac{55296}{5} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 64 a - 248\) , \( 504 a - 1952\bigr] \) |
${y}^2={x}^{3}+\left(64a-248\right){x}+504a-1952$ |
| 40.1-b4 |
40.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.638218348$ |
$32.60784567$ |
2.686678917 |
\( \frac{132304644}{5} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 217 a - 815\) , \( -3552 a + 13781\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(217a-815\right){x}-3552a+13781$ |
| 40.1-c1 |
40.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{8} \cdot 5^{8} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$4.406960782$ |
2.275744762 |
\( \frac{237276}{625} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -24 a + 107\) , \( 295 a - 1123\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-24a+107\right){x}+295a-1123$ |
| 40.1-c2 |
40.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{4} \cdot 5^{4} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$17.62784313$ |
2.275744762 |
\( \frac{148176}{25} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 16 a - 48\) , \( 37 a - 124\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(16a-48\right){x}+37a-124$ |
| 40.1-c3 |
40.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{20} \cdot 5^{2} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$35.25568626$ |
2.275744762 |
\( \frac{55296}{5} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 64 a - 248\) , \( -504 a + 1952\bigr] \) |
${y}^2={x}^{3}+\left(64a-248\right){x}-504a+1952$ |
| 40.1-c4 |
40.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$4.406960782$ |
2.275744762 |
\( \frac{132304644}{5} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 216 a - 823\) , \( 3157 a - 12209\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(216a-823\right){x}+3157a-12209$ |
| 40.1-d1 |
40.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{20} \cdot 5^{8} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$8.151961419$ |
2.104827387 |
\( \frac{237276}{625} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 13\) , \( 34\bigr] \) |
${y}^2={x}^{3}+13{x}+34$ |
| 40.1-d2 |
40.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{16} \cdot 5^{4} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$32.60784567$ |
2.104827387 |
\( \frac{148176}{25} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -7\) , \( 6\bigr] \) |
${y}^2={x}^{3}-7{x}+6$ |
| 40.1-d3 |
40.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$16.30392283$ |
2.104827387 |
\( \frac{55296}{5} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2\) , \( -1\bigr] \) |
${y}^2={x}^{3}-2{x}-1$ |
| 40.1-d4 |
40.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{20} \cdot 5^{2} \) |
$1.74072$ |
$(2,a+1), (5,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$32.60784567$ |
2.104827387 |
\( \frac{132304644}{5} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -107\) , \( 426\bigr] \) |
${y}^2={x}^{3}-107{x}+426$ |
*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.
