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 |
16.1-a5 |
16.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$0.61910$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$35.39006381$ |
0.638514464 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -a - 3\) , \( -1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-3\right){x}-1$ |
16.1-a6 |
16.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$0.61910$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$35.39006381$ |
0.638514464 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -a - 3\) , \( -a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-a-3\right){x}-a-1$ |
36.1-a5 |
36.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{6} \) |
$0.75824$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$11.79668793$ |
0.851352619 |
\( 54000 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 14 a - 26\) , \( 28 a - 50\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(14a-26\right){x}+28a-50$ |
36.1-a6 |
36.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{6} \) |
$0.75824$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$35.39006381$ |
0.851352619 |
\( 54000 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 15 a - 24\) , \( -54 a + 94\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(15a-24\right){x}-54a+94$ |
256.1-c5 |
256.1-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{16} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$17.69503190$ |
1.277028929 |
\( 54000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -4\) , \( 4 a\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}-4{x}+4a$ |
256.1-c6 |
256.1-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{16} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$17.69503190$ |
1.277028929 |
\( 54000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -4\) , \( -4 a\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}-4{x}-4a$ |
484.2-c5 |
484.2-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{6} \) |
$1.45191$ |
$(a+1), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$18.48185806$ |
1.333813215 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 5 a - 16\) , \( -14 a + 30\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(5a-16\right){x}-14a+30$ |
484.2-c6 |
484.2-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.2 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{6} \) |
$1.45191$ |
$(a+1), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$6.160619353$ |
1.333813215 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 5 a - 16\) , \( 14 a - 30\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a-16\right){x}+14a-30$ |
484.3-c5 |
484.3-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.3 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{6} \) |
$1.45191$ |
$(a+1), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$18.48185806$ |
1.333813215 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 424 a - 738\) , \( -6054 a + 10484\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(424a-738\right){x}-6054a+10484$ |
484.3-c6 |
484.3-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
484.3 |
\( 2^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{6} \) |
$1.45191$ |
$(a+1), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$6.160619353$ |
1.333813215 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 424 a - 738\) , \( 6053 a - 10486\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(424a-738\right){x}+6053a-10486$ |
676.2-a5 |
676.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.2 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a-4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.691749403$ |
$9.815437671$ |
2.396762951 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 165 a - 286\) , \( -1448 a + 2508\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(165a-286\right){x}-1448a+2508$ |
676.2-a6 |
676.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
676.2 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 13^{6} \) |
$1.57840$ |
$(a+1), (a-4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.563916467$ |
$9.815437671$ |
2.396762951 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 165 a - 286\) , \( 1448 a - 2508\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(165a-286\right){x}+1448a-2508$ |
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 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 24 a - 48\) , \( -86 a + 146\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(24a-48\right){x}-86a+146$ |
1024.1-j5 |
1024.1-j |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{22} \) |
$1.75107$ |
$(a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.859914975$ |
$21.67189957$ |
2.689873605 |
\( 54000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -10 a - 19\) , \( 31 a + 54\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-10a-19\right){x}+31a+54$ |
1024.1-j6 |
1024.1-j |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{22} \) |
$1.75107$ |
$(a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$2.579744927$ |
$7.223966526$ |
2.689873605 |
\( 54000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -10 a - 19\) , \( -31 a - 54\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-10a-19\right){x}-31a-54$ |
1024.1-k5 |
1024.1-k |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{22} \) |
$1.75107$ |
$(a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$2.579744927$ |
$7.223966526$ |
2.689873605 |
\( 54000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 10 a - 19\) , \( 31 a - 54\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(10a-19\right){x}+31a-54$ |
1024.1-k6 |
1024.1-k |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{22} \) |
$1.75107$ |
$(a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.859914975$ |
$21.67189957$ |
2.689873605 |
\( 54000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 10 a - 19\) , \( -31 a + 54\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(10a-19\right){x}-31a+54$ |
2304.1-v5 |
2304.1-v |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{6} \) |
$2.14462$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.979852016$ |
$5.898343969$ |
3.336798323 |
\( 54000 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -60 a - 105\) , \( -330 a - 572\bigr] \) |
${y}^2={x}^{3}+\left(-60a-105\right){x}-330a-572$ |
2304.1-v6 |
2304.1-v |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{6} \) |
$2.14462$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.326617338$ |
$17.69503190$ |
3.336798323 |
\( 54000 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -60 a - 105\) , \( 330 a + 572\bigr] \) |
${y}^2={x}^{3}+\left(-60a-105\right){x}+330a+572$ |
*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.