| 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 |
| 21.1-a1 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{2} \cdot 7^{16} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$7.725929285$ |
$3.651881942$ |
5.506844384 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -3167 a - 14627\) , \( 613514 a + 2836587\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3167a-14627\right){x}+613514a+2836587$ |
| 21.1-a2 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{4} \cdot 7^{2} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.965741160$ |
$14.60752776$ |
5.506844384 |
\( \frac{103823}{63} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( 88 a + 423\) , \( 272 a + 1279\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(88a+423\right){x}+272a+1279$ |
| 21.1-a3 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{8} \cdot 7^{4} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.931482321$ |
$14.60752776$ |
5.506844384 |
\( \frac{7189057}{3969} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -377 a - 1727\) , \( 1748 a + 8103\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-377a-1727\right){x}+1748a+8103$ |
| 21.1-a4 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{16} \cdot 7^{2} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.965741160$ |
$3.651881942$ |
5.506844384 |
\( \frac{6570725617}{45927} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -3632 a - 16777\) , \( -269278 a - 1244981\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3632a-16777\right){x}-269278a-1244981$ |
| 21.1-a5 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{4} \cdot 7^{8} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$3.862964642$ |
$14.60752776$ |
5.506844384 |
\( \frac{13027640977}{21609} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -4562 a - 21077\) , \( 376778 a + 1742043\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4562a-21077\right){x}+376778a+1742043$ |
| 21.1-a6 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{2} \cdot 7^{4} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$7.725929285$ |
$14.60752776$ |
5.506844384 |
\( \frac{53297461115137}{147} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -72917 a - 337127\) , \( 24227822 a + 112016739\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-72917a-337127\right){x}+24227822a+112016739$ |
| 21.1-b1 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{2} \cdot 7^{16} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.236287448$ |
$3.651881942$ |
1.347357100 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -21268357 a - 98333725\) , \( 338779129467 a + 1566336965358\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-21268357a-98333725\right){x}+338779129467a+1566336965358$ |
| 21.1-b2 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{4} \cdot 7^{2} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.890299589$ |
$14.60752776$ |
1.347357100 |
\( \frac{103823}{63} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 612138 a + 2830205\) , \( 132581164 a + 612985748\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(612138a+2830205\right){x}+132581164a+612985748$ |
| 21.1-b3 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{8} \cdot 7^{4} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.945149794$ |
$14.60752776$ |
1.347357100 |
\( \frac{7189057}{3969} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -2513647 a - 11621785\) , \( 1042518123 a + 4820056878\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2513647a-11621785\right){x}+1042518123a+4820056878$ |
| 21.1-b4 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{16} \cdot 7^{2} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.890299589$ |
$3.651881942$ |
1.347357100 |
\( \frac{6570725617}{45927} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -24394142 a - 112785715\) , \( -147650518306 a - 682658536672\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-24394142a-112785715\right){x}-147650518306a-682658536672$ |
| 21.1-b5 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{4} \cdot 7^{8} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.472574897$ |
$14.60752776$ |
1.347357100 |
\( \frac{13027640977}{21609} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -30645712 a - 141689695\) , \( 208598144508 a + 964448386068\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-30645712a-141689695\right){x}+208598144508a+964448386068$ |
| 21.1-b6 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{2} \cdot 7^{4} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.236287448$ |
$14.60752776$ |
1.347357100 |
\( \frac{53297461115137}{147} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -490136107 a - 2266132225\) , \( 13367616869409 a + 61804847524818\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-490136107a-2266132225\right){x}+13367616869409a+61804847524818$ |
| 21.1-c1 |
21.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{16} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$2.638508823$ |
$0.814020435$ |
3.353661242 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -22318 a - 103177\) , \( -11478705 a - 53071509\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-22318a-103177\right){x}-11478705a-53071509$ |
| 21.1-c2 |
21.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{2} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.319254411$ |
$13.02432697$ |
3.353661242 |
\( \frac{103823}{63} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 642 a + 2978\) , \( -5572 a - 25761\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(642a+2978\right){x}-5572a-25761$ |
| 21.1-c3 |
21.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.638508823$ |
$13.02432697$ |
3.353661242 |
\( \frac{7189057}{3969} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -2638 a - 12187\) , \( -31071 a - 143655\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-2638a-12187\right){x}-31071a-143655$ |
| 21.1-c4 |
21.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{16} \cdot 7^{2} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.319254411$ |
$13.02432697$ |
3.353661242 |
\( \frac{6570725617}{45927} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -25598 a - 118342\) , \( 5061268 a + 23400649\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-25598a-118342\right){x}+5061268a+23400649$ |
| 21.1-c5 |
21.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$5.277017646$ |
$3.256081743$ |
3.353661242 |
\( \frac{13027640977}{21609} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -32158 a - 148672\) , \( -7037346 a - 32536995\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-32158a-148672\right){x}-7037346a-32536995$ |
| 21.1-c6 |
21.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{4} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$10.55403529$ |
$0.814020435$ |
3.353661242 |
\( \frac{53297461115137}{147} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -514318 a - 2377927\) , \( -453535587 a - 2096910621\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-514318a-2377927\right){x}-453535587a-2096910621$ |
| 21.1-d1 |
21.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{16} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$7.203395932$ |
$0.814020435$ |
1.144479295 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -34\) , \( -217\bigr] \) |
${y}^2+{x}{y}={x}^{3}-34{x}-217$ |
| 21.1-d2 |
21.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{2} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.900424491$ |
$13.02432697$ |
1.144479295 |
\( \frac{103823}{63} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}$ |
| 21.1-d3 |
21.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.800848983$ |
$13.02432697$ |
1.144479295 |
\( \frac{7189057}{3969} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -4\) , \( -1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-4{x}-1$ |
| 21.1-d4 |
21.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{16} \cdot 7^{2} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.900424491$ |
$13.02432697$ |
1.144479295 |
\( \frac{6570725617}{45927} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -39\) , \( 90\bigr] \) |
${y}^2+{x}{y}={x}^{3}-39{x}+90$ |
| 21.1-d5 |
21.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$3.601697966$ |
$3.256081743$ |
1.144479295 |
\( \frac{13027640977}{21609} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -49\) , \( -136\bigr] \) |
${y}^2+{x}{y}={x}^{3}-49{x}-136$ |
| 21.1-d6 |
21.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{105}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{4} \) |
$1.96014$ |
$(3,a+1), (7,a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$7.203395932$ |
$0.814020435$ |
1.144479295 |
\( \frac{53297461115137}{147} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -784\) , \( -8515\bigr] \) |
${y}^2+{x}{y}={x}^{3}-784{x}-8515$ |
*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.
