| 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 |
| 6.1-a1 |
6.1-a |
$4$ |
$22$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{13} \cdot 3^{2} \) |
$0.95879$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 11$ |
2B, 11B.3.1 |
$1$ |
\( 2 \) |
$1$ |
$10.69327331$ |
1.559774220 |
\( \frac{6049}{18} a - \frac{5155}{18} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -4 a - 5\) , \( -3 a + 14\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+a{x}^2+\left(-4a-5\right){x}-3a+14$ |
| 6.1-a2 |
6.1-a |
$4$ |
$22$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3 \) |
$0.95879$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 11$ |
2B, 11B.3.1 |
$1$ |
\( 2 \) |
$1$ |
$10.69327331$ |
1.559774220 |
\( -\frac{12167}{12} a + \frac{12167}{12} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -4\) , \( -a + 2\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3+a{x}^2-4{x}-a+2$ |
| 6.1-a3 |
6.1-a |
$4$ |
$22$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{23} \cdot 3^{22} \) |
$0.95879$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 11$ |
2B, 11B.3.1 |
$1$ |
\( 2 \cdot 11 \) |
$1$ |
$0.972115755$ |
1.559774220 |
\( \frac{1303848704909983}{64268410079232} a + \frac{105559731166913477}{64268410079232} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -49 a + 140\) , \( 173 a + 206\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+a{x}^2+\left(-49a+140\right){x}+173a+206$ |
| 6.1-a4 |
6.1-a |
$4$ |
$22$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{22} \cdot 3^{11} \) |
$0.95879$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 11$ |
2B, 11B.3.1 |
$1$ |
\( 2 \cdot 11 \) |
$1$ |
$0.972115755$ |
1.559774220 |
\( \frac{57720999401577469}{743008370688} a + \frac{173915194158707375}{743008370688} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -30 a + 111\) , \( 96 a + 712\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3+a{x}^2+\left(-30a+111\right){x}+96a+712$ |
| 6.4-a1 |
6.4-a |
$4$ |
$22$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3 \) |
$0.95879$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 11$ |
2B, 11B.3.1 |
$1$ |
\( 2 \) |
$1$ |
$10.69327331$ |
1.559774220 |
\( \frac{12167}{12} a \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -a - 3\) , \( 2\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(-a-3\right){x}+2$ |
| 6.4-a2 |
6.4-a |
$4$ |
$22$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{13} \cdot 3^{2} \) |
$0.95879$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 11$ |
2B, 11B.3.1 |
$1$ |
\( 2 \) |
$1$ |
$10.69327331$ |
1.559774220 |
\( -\frac{6049}{18} a + \frac{149}{3} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -3 a + 4\) , \( 8\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a-1\right){x}^2+\left(-3a+4\right){x}+8$ |
| 6.4-a3 |
6.4-a |
$4$ |
$22$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{23} \cdot 3^{22} \) |
$0.95879$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 11$ |
2B, 11B.3.1 |
$1$ |
\( 2 \cdot 11 \) |
$1$ |
$0.972115755$ |
1.559774220 |
\( -\frac{1303848704909983}{64268410079232} a + \frac{8905298322651955}{5355700839936} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( 42 a + 104\) , \( -76 a - 264\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a-1\right){x}^2+\left(42a+104\right){x}-76a-264$ |
| 6.4-a4 |
6.4-a |
$4$ |
$22$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{22} \cdot 3^{11} \) |
$0.95879$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 11$ |
2B, 11B.3.1 |
$1$ |
\( 2 \cdot 11 \) |
$1$ |
$0.972115755$ |
1.559774220 |
\( -\frac{57720999401577469}{743008370688} a + \frac{19303016130023737}{61917364224} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 29 a + 82\) , \( -97 a + 809\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(29a+82\right){x}-97a+809$ |
| 12.3-a1 |
12.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
12.3 |
\( 2^{2} \cdot 3 \) |
\( 2^{24} \cdot 3^{3} \) |
$1.14020$ |
$(2,a), (2,a+1), (3,a)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$4.689167484$ |
0.455990250 |
\( \frac{60527389}{27648} a - \frac{265661425}{27648} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( a + 1\) , \( a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}+a+2$ |
| 12.3-a2 |
12.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
12.3 |
\( 2^{2} \cdot 3 \) |
\( 2^{9} \cdot 3^{18} \) |
$1.14020$ |
$(2,a), (2,a+1), (3,a)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$4.689167484$ |
0.455990250 |
\( \frac{1687537}{23328} a - \frac{17625427}{23328} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( a - 18\) , \( -36\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(a-1\right){x}^2+\left(a-18\right){x}-36$ |
| 12.3-a3 |
12.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
12.3 |
\( 2^{2} \cdot 3 \) |
\( 2^{48} \cdot 3 \) |
$1.14020$ |
$(2,a), (2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1.563055828$ |
0.455990250 |
\( -\frac{5581037003}{3221225472} a + \frac{6004501103255}{3221225472} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -24 a + 46\) , \( 19 a + 104\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-24a+46\right){x}+19a+104$ |
| 12.3-a4 |
12.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
12.3 |
\( 2^{2} \cdot 3 \) |
\( 2^{27} \cdot 3^{14} \) |
$1.14020$ |
$(2,a), (2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1.563055828$ |
0.455990250 |
\( -\frac{1328949641}{294912} a + \frac{218838131861}{294912} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -159 a - 33\) , \( 1265 a - 2355\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(a-1\right){x}^2+\left(-159a-33\right){x}+1265a-2355$ |
| 12.4-a1 |
12.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
12.4 |
\( 2^{2} \cdot 3 \) |
\( 2^{24} \cdot 3^{3} \) |
$1.14020$ |
$(2,a), (2,a+1), (3,a+2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$4.689167484$ |
0.455990250 |
\( -\frac{60527389}{27648} a - \frac{17094503}{2304} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -3 a + 4\) , \( -2 a + 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(-3a+4\right){x}-2a+4$ |
| 12.4-a2 |
12.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
12.4 |
\( 2^{2} \cdot 3 \) |
\( 2^{9} \cdot 3^{18} \) |
$1.14020$ |
$(2,a), (2,a+1), (3,a+2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$4.689167484$ |
0.455990250 |
\( -\frac{1687537}{23328} a - \frac{2656315}{3888} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -2 a - 17\) , \( -a - 36\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-2a-17\right){x}-a-36$ |
| 12.4-a3 |
12.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
12.4 |
\( 2^{2} \cdot 3 \) |
\( 2^{48} \cdot 3 \) |
$1.14020$ |
$(2,a), (2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1.563055828$ |
0.455990250 |
\( \frac{5581037003}{3221225472} a + \frac{499910005521}{268435456} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 22 a + 24\) , \( -20 a + 124\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(22a+24\right){x}-20a+124$ |
| 12.4-a4 |
12.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
12.4 |
\( 2^{2} \cdot 3 \) |
\( 2^{27} \cdot 3^{14} \) |
$1.14020$ |
$(2,a), (2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1.563055828$ |
0.455990250 |
\( \frac{1328949641}{294912} a + \frac{18125765185}{24576} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 158 a - 192\) , \( -1266 a - 1090\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(158a-192\right){x}-1266a-1090$ |
| 17.1-a1 |
17.1-a |
$2$ |
$11$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
17.1 |
\( 17 \) |
\( 2^{12} \cdot 17 \) |
$1.24394$ |
$(17,a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$11$ |
11B.3.1 |
$1$ |
\( 1 \) |
$0.191050476$ |
$11.03460706$ |
1.230031008 |
\( \frac{67}{17} a + \frac{2065}{17} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -2 a + 4\) , \( -a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+{x}^2+\left(-2a+4\right){x}-a+4$ |
| 17.1-a2 |
17.1-a |
$2$ |
$11$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
17.1 |
\( 17 \) |
\( 2^{12} \cdot 17^{11} \) |
$1.24394$ |
$(17,a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$11$ |
11B.3.1 |
$1$ |
\( 1 \) |
$2.101555239$ |
$1.003146096$ |
1.230031008 |
\( -\frac{5829173027496149}{34271896307633} a + \frac{91145922407056009}{34271896307633} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -12 a + 214\) , \( 245 a + 118\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+{x}^2+\left(-12a+214\right){x}+245a+118$ |
| 17.2-a1 |
17.2-a |
$2$ |
$11$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
17.2 |
\( 17 \) |
\( 2^{12} \cdot 17 \) |
$1.24394$ |
$(17,a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$11$ |
11B.3.1 |
$1$ |
\( 1 \) |
$0.191050476$ |
$11.03460706$ |
1.230031008 |
\( -\frac{67}{17} a + \frac{2132}{17} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 2 a + 3\) , \( -a + 1\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(2a+3\right){x}-a+1$ |
| 17.2-a2 |
17.2-a |
$2$ |
$11$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
17.2 |
\( 17 \) |
\( 2^{12} \cdot 17^{11} \) |
$1.24394$ |
$(17,a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$11$ |
11B.3.1 |
$1$ |
\( 1 \) |
$2.101555239$ |
$1.003146096$ |
1.230031008 |
\( \frac{5829173027496149}{34271896307633} a + \frac{85316749379559860}{34271896307633} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 12 a + 203\) , \( -257 a + 161\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(12a+203\right){x}-257a+161$ |
| 27.2-a1 |
27.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{24} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$4.573570113$ |
0.667123765 |
\( \frac{5096}{81} a + \frac{136489}{81} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 4 a - 23\) , \( 5 a + 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(4a-23\right){x}+5a+6$ |
| 27.2-a2 |
27.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{12} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$4.573570113$ |
0.667123765 |
\( -\frac{5096}{81} a + \frac{47195}{27} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -a + 19\) , \( -a - 8\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-a+19\right){x}-a-8$ |
| 27.2-a3 |
27.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{9} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$4.573570113$ |
0.667123765 |
\( \frac{230594}{9} a + 77433 \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( a + 1\) , \( -3 a + 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(a+1\right){x}-3a+2$ |
| 27.2-a4 |
27.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{21} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$4.573570113$ |
0.667123765 |
\( -\frac{230594}{9} a + \frac{927491}{9} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -2 a + 49\) , \( -38 a - 12\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(-2a+49\right){x}-38a-12$ |
| 27.2-a5 |
27.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{27} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.286785056$ |
0.667123765 |
\( \frac{971370218}{6561} a + \frac{2242143013}{6561} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 34 a - 248\) , \( 353 a - 1146\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(34a-248\right){x}+353a-1146$ |
| 27.2-a6 |
27.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{15} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.286785056$ |
0.667123765 |
\( -\frac{971370218}{6561} a + \frac{1071171077}{2187} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -6 a + 124\) , \( 127 a - 104\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-6a+124\right){x}+127a-104$ |
| 27.2-b1 |
27.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{16} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$0.482148235$ |
$3.136639768$ |
1.764762571 |
\( \frac{3899392}{19683} a - \frac{18448384}{19683} \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( 5 a - 6\) , \( 9 a + 1\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+a{x}^2+\left(5a-6\right){x}+9a+1$ |
| 27.2-b2 |
27.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{28} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.482148235$ |
$3.136639768$ |
1.764762571 |
\( -\frac{3899392}{19683} a - \frac{4849664}{6561} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 9 a + 12\) , \( 20 a - 165\bigr] \) |
${y}^2+{y}={x}^3+a{x}^2+\left(9a+12\right){x}+20a-165$ |
| 27.2-b3 |
27.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{12} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1.446444706$ |
$3.136639768$ |
1.764762571 |
\( \frac{11239424}{27} \) |
\( \bigl[0\) , \( a\) , \( a\) , \( 5 a - 60\) , \( 2 a - 172\bigr] \) |
${y}^2+a{y}={x}^3+a{x}^2+\left(5a-60\right){x}+2a-172$ |
| 27.2-c1 |
27.2-c |
$2$ |
$17$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{24} \cdot 17^{12} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$17$ |
17B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.289909742$ |
1.505221389 |
\( \frac{1144804950016}{129140163} a - \frac{589982863360}{129140163} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -2985 a + 12753\) , \( -103117 a - 599175\bigr] \) |
${y}^2+{y}={x}^3-a{x}^2+\left(-2985a+12753\right){x}-103117a-599175$ |
| 27.2-c2 |
27.2-c |
$2$ |
$17$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{24} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$17$ |
17B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.289909742$ |
1.505221389 |
\( -\frac{1144804950016}{129140163} a + \frac{184940695552}{43046721} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 15 a - 27\) , \( -37 a + 27\bigr] \) |
${y}^2+{y}={x}^3-a{x}^2+\left(15a-27\right){x}-37a+27$ |
| 27.3-a1 |
27.3-a |
$2$ |
$17$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{24} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$17$ |
17B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.289909742$ |
1.505221389 |
\( \frac{1144804950016}{129140163} a - \frac{589982863360}{129140163} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -15 a - 12\) , \( 37 a - 10\bigr] \) |
${y}^2+{y}={x}^3+\left(a-1\right){x}^2+\left(-15a-12\right){x}+37a-10$ |
| 27.3-a2 |
27.3-a |
$2$ |
$17$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{36} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$17$ |
17B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.289909742$ |
1.505221389 |
\( -\frac{1144804950016}{129140163} a + \frac{184940695552}{43046721} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 132 a - 207\) , \( 1085 a + 860\bigr] \) |
${y}^2+{y}={x}^3+\left(132a-207\right){x}+1085a+860$ |
| 27.3-b1 |
27.3-b |
$3$ |
$9$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{28} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.482148235$ |
$3.136639768$ |
1.764762571 |
\( \frac{3899392}{19683} a - \frac{18448384}{19683} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -9 a + 21\) , \( -20 a - 145\bigr] \) |
${y}^2+{y}={x}^3+\left(-a+1\right){x}^2+\left(-9a+21\right){x}-20a-145$ |
| 27.3-b2 |
27.3-b |
$3$ |
$9$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{16} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$0.482148235$ |
$3.136639768$ |
1.764762571 |
\( -\frac{3899392}{19683} a - \frac{4849664}{6561} \) |
\( \bigl[0\) , \( -a + 1\) , \( a\) , \( -5 a - 1\) , \( -10 a + 11\bigr] \) |
${y}^2+a{y}={x}^3+\left(-a+1\right){x}^2+\left(-5a-1\right){x}-10a+11$ |
| 27.3-b3 |
27.3-b |
$3$ |
$9$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{12} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.446444706$ |
$3.136639768$ |
1.764762571 |
\( \frac{11239424}{27} \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( -5 a - 55\) , \( -3 a - 170\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(-5a-55\right){x}-3a-170$ |
| 27.3-c1 |
27.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{12} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$4.573570113$ |
0.667123765 |
\( \frac{5096}{81} a + \frac{136489}{81} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -a + 20\) , \( -8\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(-a+20\right){x}-8$ |
| 27.3-c2 |
27.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{24} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$4.573570113$ |
0.667123765 |
\( -\frac{5096}{81} a + \frac{47195}{27} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -2 a - 19\) , \( -8 a - 8\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(-2a-19\right){x}-8a-8$ |
| 27.3-c3 |
27.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{21} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$4.573570113$ |
0.667123765 |
\( \frac{230594}{9} a + 77433 \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( a + 48\) , \( 37 a - 49\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3-{x}^2+\left(a+48\right){x}+37a-49$ |
| 27.3-c4 |
27.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{9} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$4.573570113$ |
0.667123765 |
\( -\frac{230594}{9} a + \frac{927491}{9} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( a + 2\) , \( 3 a + 1\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(a+2\right){x}+3a+1$ |
| 27.3-c5 |
27.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{15} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.286785056$ |
0.667123765 |
\( \frac{971370218}{6561} a + \frac{2242143013}{6561} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 4 a + 120\) , \( -128 a + 24\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(4a+120\right){x}-128a+24$ |
| 27.3-c6 |
27.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{27} \) |
$1.39646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.286785056$ |
0.667123765 |
\( -\frac{971370218}{6561} a + \frac{1071171077}{2187} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -32 a - 214\) , \( -386 a - 1007\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(-32a-214\right){x}-386a-1007$ |
| 32.2-a1 |
32.2-a |
$4$ |
$14$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{27} \) |
$1.45705$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.934093823$ |
0.855963141 |
\( \frac{582471}{16384} a + \frac{181629}{16384} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( a + 1\) , \( -6\bigr] \) |
${y}^2+a{x}{y}={x}^3-a{x}^2+\left(a+1\right){x}-6$ |
| 32.2-a2 |
32.2-a |
$4$ |
$14$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{27} \cdot 3^{12} \) |
$1.45705$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.934093823$ |
0.855963141 |
\( -\frac{582471}{16384} a + \frac{191025}{4096} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( 7 a + 9\) , \( 22 a - 94\bigr] \) |
${y}^2+a{x}{y}={x}^3-a{x}^2+\left(7a+9\right){x}+22a-94$ |
| 32.2-a3 |
32.2-a |
$4$ |
$14$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{33} \) |
$1.45705$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.934093823$ |
0.855963141 |
\( \frac{8142201}{128} a + \frac{2318841}{32} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 16 a + 13\) , \( 12 a + 162\bigr] \) |
${y}^2={x}^3+\left(16a+13\right){x}+12a+162$ |
| 32.2-a4 |
32.2-a |
$4$ |
$14$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{33} \) |
$1.45705$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.934093823$ |
0.855963141 |
\( -\frac{8142201}{128} a + \frac{17417565}{128} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -16 a + 29\) , \( 12 a - 174\bigr] \) |
${y}^2={x}^3+\left(-16a+29\right){x}+12a-174$ |
| 32.5-a1 |
32.5-a |
$4$ |
$14$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{27} \cdot 3^{12} \) |
$1.45705$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.934093823$ |
0.855963141 |
\( \frac{582471}{16384} a + \frac{181629}{16384} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -7 a + 16\) , \( -22 a - 72\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3-{x}^2+\left(-7a+16\right){x}-22a-72$ |
| 32.5-a2 |
32.5-a |
$4$ |
$14$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{27} \) |
$1.45705$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.934093823$ |
0.855963141 |
\( -\frac{582471}{16384} a + \frac{191025}{4096} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -a + 2\) , \( -6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3-{x}^2+\left(-a+2\right){x}-6$ |
| 32.5-a3 |
32.5-a |
$4$ |
$14$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{33} \) |
$1.45705$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.934093823$ |
0.855963141 |
\( \frac{8142201}{128} a + \frac{2318841}{32} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 16 a + 13\) , \( -12 a - 162\bigr] \) |
${y}^2={x}^3+\left(16a+13\right){x}-12a-162$ |
| 32.5-a4 |
32.5-a |
$4$ |
$14$ |
\(\Q(\sqrt{-47}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{33} \) |
$1.45705$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 7$ |
2B, 7B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.934093823$ |
0.855963141 |
\( -\frac{8142201}{128} a + \frac{17417565}{128} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -16 a + 29\) , \( -12 a + 174\bigr] \) |
${y}^2={x}^3+\left(-16a+29\right){x}-12a+174$ |
*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.
