| 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 |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{2} \cdot 7^{16} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$20.59050663$ |
$0.814020435$ |
5.172585655 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -4769455 a - 30909537\) , \( -40477455354 a - 262323892140\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4769455a-30909537\right){x}-40477455354a-262323892140$ |
| 21.1-a2 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{4} \cdot 7^{2} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.286906664$ |
$13.02432697$ |
5.172585655 |
\( \frac{103823}{63} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( 137265 a + 889643\) , \( -15391334 a - 99747104\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(137265a+889643\right){x}-15391334a-99747104$ |
| 21.1-a3 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{8} \cdot 7^{4} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$5.147626658$ |
$13.02432697$ |
5.172585655 |
\( \frac{7189057}{3969} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -563695 a - 3653097\) , \( -126334194 a - 818739012\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-563695a-3653097\right){x}-126334194a-818739012$ |
| 21.1-a4 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{16} \cdot 7^{2} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.286906664$ |
$13.02432697$ |
5.172585655 |
\( \frac{6570725617}{45927} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -5470415 a - 35452277\) , \( 17616948986 a + 114170878416\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5470415a-35452277\right){x}+17616948986a+114170878416$ |
| 21.1-a5 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{4} \cdot 7^{8} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$20.59050663$ |
$3.256081743$ |
5.172585655 |
\( \frac{13027640977}{21609} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -6872335 a - 44537757\) , \( -24936098334 a - 161604387192\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6872335a-44537757\right){x}-24936098334a-161604387192$ |
| 21.1-a6 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{2} \cdot 7^{4} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$20.59050663$ |
$0.814020435$ |
5.172585655 |
\( \frac{53297461115137}{147} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -109913455 a - 712320537\) , \( -1596914887554 a - 10349191303524\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-109913455a-712320537\right){x}-1596914887554a-10349191303524$ |
| 21.1-b1 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{2} \cdot 7^{16} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$8.753013071$ |
$3.651881942$ |
4.932302010 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -7084 a - 45802\) , \( 2301132 a + 14913254\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-7084a-45802\right){x}+2301132a+14913254$ |
| 21.1-b2 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{4} \cdot 7^{2} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.094126633$ |
$14.60752776$ |
4.932302010 |
\( \frac{103823}{63} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 196 a + 1378\) , \( 1212 a + 8070\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(196a+1378\right){x}+1212a+8070$ |
| 21.1-b3 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{8} \cdot 7^{4} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$2.188253267$ |
$14.60752776$ |
4.932302010 |
\( \frac{7189057}{3969} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -844 a - 5362\) , \( 5772 a + 37622\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-844a-5362\right){x}+5772a+37622$ |
| 21.1-b4 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{16} \cdot 7^{2} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1.094126633$ |
$3.651881942$ |
4.932302010 |
\( \frac{6570725617}{45927} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -8124 a - 52542\) , \( -1020708 a - 6614730\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8124a-52542\right){x}-1020708a-6614730$ |
| 21.1-b5 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{4} \cdot 7^{8} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$4.376506535$ |
$14.60752776$ |
4.932302010 |
\( \frac{13027640977}{21609} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -10204 a - 66022\) , \( 1407612 a + 9122582\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-10204a-66022\right){x}+1407612a+9122582$ |
| 21.1-b6 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 2^{12} \cdot 3^{2} \cdot 7^{4} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$8.753013071$ |
$14.60752776$ |
4.932302010 |
\( \frac{53297461115137}{147} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -163084 a - 1056802\) , \( 90983532 a + 589640870\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-163084a-1056802\right){x}+90983532a+589640870$ |
| 21.1-c1 |
21.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{16} \) |
$2.47941$ |
$(3,a), (a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$3.651881942$ |
1.126995234 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 1772 a - 11437\) , \( -284098 a + 1841276\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(1772a-11437\right){x}-284098a+1841276$ |
| 21.1-c2 |
21.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{2} \) |
$2.47941$ |
$(3,a), (a+7)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$14.60752776$ |
1.126995234 |
\( \frac{103823}{63} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -48 a + 358\) , \( -248 a + 1718\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-48a+358\right){x}-248a+1718$ |
| 21.1-c3 |
21.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$2.47941$ |
$(3,a), (a+7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$14.60752776$ |
1.126995234 |
\( \frac{7189057}{3969} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 212 a - 1327\) , \( -298 a + 2042\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(212a-1327\right){x}-298a+2042$ |
| 21.1-c4 |
21.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{16} \cdot 7^{2} \) |
$2.47941$ |
$(3,a), (a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$3.651881942$ |
1.126995234 |
\( \frac{6570725617}{45927} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 2032 a - 13122\) , \( 131652 a - 853092\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(2032a-13122\right){x}+131652a-853092$ |
| 21.1-c5 |
21.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$2.47941$ |
$(3,a), (a+7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$14.60752776$ |
1.126995234 |
\( \frac{13027640977}{21609} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 2552 a - 16492\) , \( -170848 a + 1107332\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(2552a-16492\right){x}-170848a+1107332$ |
| 21.1-c6 |
21.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{4} \) |
$2.47941$ |
$(3,a), (a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$14.60752776$ |
1.126995234 |
\( \frac{53297461115137}{147} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 40772 a - 264187\) , \( -11291398 a + 73176728\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(40772a-264187\right){x}-11291398a+73176728$ |
| 21.1-d1 |
21.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{16} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$23.79400264$ |
$0.814020435$ |
2.988671404 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -34\) , \( -217\bigr] \) |
${y}^2+{x}{y}={x}^{3}-34{x}-217$ |
| 21.1-d2 |
21.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{2} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.974250330$ |
$13.02432697$ |
2.988671404 |
\( \frac{103823}{63} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}$ |
| 21.1-d3 |
21.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$5.948500660$ |
$13.02432697$ |
2.988671404 |
\( \frac{7189057}{3969} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -4\) , \( -1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-4{x}-1$ |
| 21.1-d4 |
21.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{16} \cdot 7^{2} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$2.974250330$ |
$13.02432697$ |
2.988671404 |
\( \frac{6570725617}{45927} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -39\) , \( 90\bigr] \) |
${y}^2+{x}{y}={x}^{3}-39{x}+90$ |
| 21.1-d5 |
21.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$11.89700132$ |
$3.256081743$ |
2.988671404 |
\( \frac{13027640977}{21609} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -49\) , \( -136\bigr] \) |
${y}^2+{x}{y}={x}^{3}-49{x}-136$ |
| 21.1-d6 |
21.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{4} \) |
$2.47941$ |
$(3,a), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$23.79400264$ |
$0.814020435$ |
2.988671404 |
\( \frac{53297461115137}{147} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -784\) , \( -8515\bigr] \) |
${y}^2+{x}{y}={x}^{3}-784{x}-8515$ |
*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.
