| 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 |
| 676.3-a1 |
676.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.845874701$ |
$4.907718835$ |
2.396762951 |
\( 0 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 1\) , \( 11 a + 17\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+{x}+11a+17$ |
| 676.3-a2 |
676.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.281958233$ |
$4.907718835$ |
2.396762951 |
\( 0 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 1\) , \( -11 a - 17\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+{x}-11a-17$ |
| 676.3-a3 |
676.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$3.383498806$ |
$2.453859417$ |
2.396762951 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 409 a - 718\) , \( 5865 a - 10165\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(409a-718\right){x}+5865a-10165$ |
| 676.3-a4 |
676.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.281958233$ |
$9.815437671$ |
2.396762951 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 409 a - 718\) , \( -5866 a + 10163\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(409a-718\right){x}-5866a+10163$ |
| 676.3-a5 |
676.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a+4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.691749403$ |
$9.815437671$ |
2.396762951 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 24 a - 48\) , \( 85 a - 148\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(24a-48\right){x}+85a-148$ |
| 676.3-a6 |
676.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a+4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.563916467$ |
$9.815437671$ |
2.396762951 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 24 a - 48\) , \( -86 a + 146\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(24a-48\right){x}-86a+146$ |
| 676.3-a7 |
676.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.845874701$ |
$9.815437671$ |
2.396762951 |
\( 818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 39 a - 118\) , \( -125 a + 325\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(39a-118\right){x}-125a+325$ |
| 676.3-a8 |
676.3-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \cdot 3 \) |
$1.127832935$ |
$2.453859417$ |
2.396762951 |
\( 818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 39 a - 118\) , \( 124 a - 327\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(39a-118\right){x}+124a-327$ |
| 676.3-b1 |
676.3-b |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 13^{2} \) |
$1.57840$ |
$(a+1), (a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.283132464$ |
$14.53909775$ |
2.376656948 |
\( 0 \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( 1\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+{x}$ |
| 676.3-b2 |
676.3-b |
$2$ |
$3$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.3 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 13^{2} \) |
$1.57840$ |
$(a+1), (a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$0.094377488$ |
$14.53909775$ |
2.376656948 |
\( 0 \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( 1\) , \( -a - 2\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+a{x}^{2}+{x}-a-2$ |
*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.
