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 |
5.1-a1 |
5.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( - 2^{12} \cdot 5^{3} \) |
$1.36923$ |
$(2a-11)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 3 \) |
$1.378497038$ |
$33.76254038$ |
1.703246763 |
\( -\frac{1145960298}{25} a + \frac{6444279999}{25} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 101 a + 507\) , \( 12625 a + 58420\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(101a+507\right){x}+12625a+58420$ |
5.1-a2 |
5.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{12} \) |
$1.36923$ |
$(2a-11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.344624259$ |
$8.440635095$ |
1.703246763 |
\( \frac{55306341}{15625} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -426911 a - 1973810\) , \( -245763021 a - 1136279283\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-426911a-1973810\right){x}-245763021a-1136279283$ |
5.1-a3 |
5.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{6} \) |
$1.36923$ |
$(2a-11)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Nn |
$1$ |
\( 2 \cdot 3 \) |
$0.689248519$ |
$33.76254038$ |
1.703246763 |
\( \frac{2803221}{125} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -157991 a - 730465\) , \( 74372212 a + 343858086\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-157991a-730465\right){x}+74372212a+343858086$ |
5.1-a4 |
5.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( - 2^{12} \cdot 5^{3} \) |
$1.36923$ |
$(2a-11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 3 \) |
$0.344624259$ |
$33.76254038$ |
1.703246763 |
\( \frac{1145960298}{25} a + \frac{5298319701}{25} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -2500155 a - 11559365\) , \( 4867995135 a + 22507055720\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2500155a-11559365\right){x}+4867995135a+22507055720$ |
5.1-b1 |
5.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{3} \) |
$1.36923$ |
$(2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$4$ |
\( 1 \) |
$1$ |
$6.686095992$ |
0.652496156 |
\( -\frac{1145960298}{25} a + \frac{6444279999}{25} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -2\) , \( -a - 10\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}-2{x}-a-10$ |
5.1-b2 |
5.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 2^{12} \cdot 5^{12} \) |
$1.36923$ |
$(2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2 \) |
$1$ |
$13.37219198$ |
0.652496156 |
\( \frac{55306341}{15625} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -12 a - 37\) , \( 12 a + 69\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-12a-37\right){x}+12a+69$ |
5.1-b3 |
5.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 2^{12} \cdot 5^{6} \) |
$1.36923$ |
$(2a-11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Nn |
$4$ |
\( 2 \) |
$1$ |
$13.37219198$ |
0.652496156 |
\( \frac{2803221}{125} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -7 a - 7\) , \( -14 a - 39\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-7a-7\right){x}-14a-39$ |
5.1-b4 |
5.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{3} \) |
$1.36923$ |
$(2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$4$ |
\( 1 \) |
$1$ |
$6.686095992$ |
0.652496156 |
\( \frac{1145960298}{25} a + \frac{5298319701}{25} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -2624 a - 12130\) , \( -164909 a - 762456\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-2624a-12130\right){x}-164909a-762456$ |
5.1-c1 |
5.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{3} \) |
$1.36923$ |
$(2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$4$ |
\( 1 \) |
$1$ |
$6.686095992$ |
0.652496156 |
\( -\frac{1145960298}{25} a + \frac{6444279999}{25} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 2623 a - 14753\) , \( 164908 a - 927364\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(2623a-14753\right){x}+164908a-927364$ |
5.1-c2 |
5.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 2^{12} \cdot 5^{12} \) |
$1.36923$ |
$(2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2 \) |
$1$ |
$13.37219198$ |
0.652496156 |
\( \frac{55306341}{15625} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 7164 a - 40283\) , \( -508682 a + 2860562\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(7164a-40283\right){x}-508682a+2860562$ |
5.1-c3 |
5.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 2^{12} \cdot 5^{6} \) |
$1.36923$ |
$(2a-11)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Nn |
$4$ |
\( 2 \) |
$1$ |
$13.37219198$ |
0.652496156 |
\( \frac{2803221}{125} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 2649 a - 14893\) , \( 171401 a - 963868\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2649a-14893\right){x}+171401a-963868$ |
5.1-c4 |
5.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{3} \) |
$1.36923$ |
$(2a-11)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$4$ |
\( 1 \) |
$1$ |
$6.686095992$ |
0.652496156 |
\( \frac{1145960298}{25} a + \frac{5298319701}{25} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -a - 1\) , \( -10\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-a-1\right){x}-10$ |
5.1-d1 |
5.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( - 2^{12} \cdot 5^{3} \) |
$1.36923$ |
$(2a-11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 3 \) |
$0.344624259$ |
$33.76254038$ |
1.703246763 |
\( -\frac{1145960298}{25} a + \frac{6444279999}{25} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 40 a - 237\) , \( -204 a + 1141\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(40a-237\right){x}-204a+1141$ |
5.1-d2 |
5.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{12} \) |
$1.36923$ |
$(2a-11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.344624259$ |
$8.440635095$ |
1.703246763 |
\( \frac{55306341}{15625} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -64 a - 294\) , \( -431 a - 1996\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-64a-294\right){x}-431a-1996$ |
5.1-d3 |
5.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{6} \) |
$1.36923$ |
$(2a-11)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3Nn |
$1$ |
\( 2 \cdot 3 \) |
$0.689248519$ |
$33.76254038$ |
1.703246763 |
\( \frac{2803221}{125} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -24 a - 109\) , \( 140 a + 644\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-24a-109\right){x}+140a+644$ |
5.1-d4 |
5.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( - 2^{12} \cdot 5^{3} \) |
$1.36923$ |
$(2a-11)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 3 \) |
$1.378497038$ |
$33.76254038$ |
1.703246763 |
\( \frac{1145960298}{25} a + \frac{5298319701}{25} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -376 a - 1709\) , \( 8426 a + 39001\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-376a-1709\right){x}+8426a+39001$ |
*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.