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 |
4.2-a1 |
4.2-a |
$6$ |
$45$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
4.2 |
\( 2^{2} \) |
\( 2^{42} \) |
$0.78920$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs.1.1, 5B |
$1$ |
\( 3 \cdot 5 \) |
$1$ |
$1.974258651$ |
1.053781309 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 22 a - 81\) , \( -177 a + 91\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(22a-81\right){x}-177a+91$ |
4.2-a2 |
4.2-a |
$6$ |
$45$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
4.2 |
\( 2^{2} \) |
\( 2^{62} \) |
$0.78920$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs.1.1, 5B |
$1$ |
\( 3^{2} \cdot 5 \) |
$1$ |
$0.658086217$ |
1.053781309 |
\( \frac{60355066783497695}{35184372088832} a - \frac{404225426482164793}{17592186044416} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -58 a - 721\) , \( -1313 a - 6309\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-58a-721\right){x}-1313a-6309$ |
4.2-a3 |
4.2-a |
$6$ |
$45$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
4.2 |
\( 2^{2} \) |
\( 2^{62} \) |
$0.78920$ |
$(2,a), (2,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs.1.1, 5B |
$1$ |
\( 5 \) |
$1$ |
$0.658086217$ |
1.053781309 |
\( -\frac{60355066783497695}{35184372088832} a - \frac{748095786180831891}{35184372088832} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 217 a + 214\) , \( 296 a + 6688\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(217a+214\right){x}+296a+6688$ |
4.2-a4 |
4.2-a |
$6$ |
$45$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
4.2 |
\( 2^{2} \) |
\( 2^{18} \) |
$0.78920$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs.1.1, 5B |
$1$ |
\( 3 \) |
$1$ |
$9.871293256$ |
1.053781309 |
\( \frac{1331}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -3 a - 6\) , \( 8\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-3a-6\right){x}+8$ |
4.2-a5 |
4.2-a |
$6$ |
$45$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
4.2 |
\( 2^{2} \) |
\( 2^{22} \) |
$0.78920$ |
$(2,a), (2,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs.1.1, 5B |
$1$ |
\( 1 \) |
$1$ |
$3.290431085$ |
1.053781309 |
\( \frac{2734106225}{512} a + \frac{20342826109}{512} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -8 a + 89\) , \( 167 a - 29\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-8a+89\right){x}+167a-29$ |
4.2-a6 |
4.2-a |
$6$ |
$45$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
4.2 |
\( 2^{2} \) |
\( 2^{22} \) |
$0.78920$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs.1.1, 5B |
$1$ |
\( 3^{2} \) |
$1$ |
$3.290431085$ |
1.053781309 |
\( -\frac{2734106225}{512} a + \frac{11538466167}{256} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -33 a + 4\) , \( 106 a - 102\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-33a+4\right){x}+106a-102$ |
4.2-b1 |
4.2-b |
$6$ |
$45$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
4.2 |
\( 2^{2} \) |
\( 2^{42} \) |
$0.78920$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs.1.1, 5B |
$1$ |
\( 3 \cdot 5 \) |
$1$ |
$1.974258651$ |
1.053781309 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -26 a - 49\) , \( 101 a + 159\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-26a-49\right){x}+101a+159$ |
4.2-b2 |
4.2-b |
$6$ |
$45$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
4.2 |
\( 2^{2} \) |
\( 2^{62} \) |
$0.78920$ |
$(2,a), (2,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs.1.1, 5B |
$1$ |
\( 5 \) |
$1$ |
$0.658086217$ |
1.053781309 |
\( \frac{60355066783497695}{35184372088832} a - \frac{404225426482164793}{17592186044416} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -221 a + 441\) , \( -77 a + 9179\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-221a+441\right){x}-77a+9179$ |
4.2-b3 |
4.2-b |
$6$ |
$45$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
4.2 |
\( 2^{2} \) |
\( 2^{62} \) |
$0.78920$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs.1.1, 5B |
$1$ |
\( 3^{2} \cdot 5 \) |
$1$ |
$0.658086217$ |
1.053781309 |
\( -\frac{60355066783497695}{35184372088832} a - \frac{748095786180831891}{35184372088832} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 54 a - 769\) , \( 597 a - 8177\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(54a-769\right){x}+597a-8177$ |
4.2-b4 |
4.2-b |
$6$ |
$45$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
4.2 |
\( 2^{2} \) |
\( 2^{18} \) |
$0.78920$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs.1.1, 5B |
$1$ |
\( 3 \) |
$1$ |
$9.871293256$ |
1.053781309 |
\( \frac{1331}{8} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -a + 1\) , \( -a + 3\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-a+1\right){x}-a+3$ |
4.2-b5 |
4.2-b |
$6$ |
$45$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
4.2 |
\( 2^{2} \) |
\( 2^{22} \) |
$0.78920$ |
$(2,a), (2,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs.1.1, 5B |
$1$ |
\( 3^{2} \) |
$1$ |
$3.290431085$ |
1.053781309 |
\( \frac{2734106225}{512} a + \frac{20342826109}{512} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 29 a - 19\) , \( -97 a - 301\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(29a-19\right){x}-97a-301$ |
4.2-b6 |
4.2-b |
$6$ |
$45$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
4.2 |
\( 2^{2} \) |
\( 2^{22} \) |
$0.78920$ |
$(2,a), (2,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3Cs.1.1, 5B |
$1$ |
\( 1 \) |
$1$ |
$3.290431085$ |
1.053781309 |
\( -\frac{2734106225}{512} a + \frac{11538466167}{256} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 4 a + 91\) , \( -73 a + 83\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(4a+91\right){x}-73a+83$ |
12.2-a1 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{52} \cdot 3^{2} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.694620345$ |
1.112282735 |
\( -\frac{4395631034341}{3145728} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -342 a - 677\) , \( 6146 a + 1975\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(-342a-677\right){x}+6146a+1975$ |
12.2-a2 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{50} \cdot 3 \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.694620345$ |
1.112282735 |
\( \frac{1073716146673271}{3298534883328} a + \frac{933287514135199}{1649267441664} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -18 a - 61\) , \( 84 a - 35\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+\left(-18a-61\right){x}+84a-35$ |
12.2-a3 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{50} \cdot 3^{13} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.694620345$ |
1.112282735 |
\( -\frac{1073716146673271}{3298534883328} a + \frac{2940291174943669}{3298534883328} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 160 a - 681\) , \( 2798 a + 287\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3-{x}^2+\left(160a-681\right){x}+2798a+287$ |
12.2-a4 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{20} \cdot 3^{10} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.473101729$ |
1.112282735 |
\( \frac{5735339}{3888} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 3 a + 13\) , \( -4 a - 5\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(3a+13\right){x}-4a-5$ |
12.2-a5 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{16} \cdot 3^{20} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.736550864$ |
1.112282735 |
\( \frac{476379541}{236196} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -17 a - 27\) , \( -20 a + 27\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(-17a-27\right){x}-20a+27$ |
12.2-a6 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{10} \cdot 3^{17} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.473101729$ |
1.112282735 |
\( \frac{572452561}{6912} a + \frac{201107101}{6912} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -20 a + 84\) , \( -55 a - 298\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3-{x}^2+\left(-20a+84\right){x}-55a-298$ |
12.2-a7 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{10} \cdot 3^{5} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.473101729$ |
1.112282735 |
\( -\frac{572452561}{6912} a + \frac{386779831}{3456} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 2 a + 4\) , \( 4\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+\left(2a+4\right){x}+4$ |
12.2-a8 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{32} \cdot 3^{4} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{3} \cdot 5 \) |
$1$ |
$0.347310172$ |
1.112282735 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -5462 a - 10917\) , \( 389122 a + 96183\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(-5462a-10917\right){x}+389122a+96183$ |
12.2-b1 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{52} \cdot 3^{2} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.694620345$ |
1.112282735 |
\( -\frac{4395631034341}{3145728} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 340 a - 1019\) , \( -6147 a + 8121\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(340a-1019\right){x}-6147a+8121$ |
12.2-b2 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{50} \cdot 3^{13} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.694620345$ |
1.112282735 |
\( \frac{1073716146673271}{3298534883328} a + \frac{933287514135199}{1649267441664} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -161 a - 521\) , \( -2799 a + 3085\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(-161a-521\right){x}-2799a+3085$ |
12.2-b3 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{50} \cdot 3 \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.694620345$ |
1.112282735 |
\( -\frac{1073716146673271}{3298534883328} a + \frac{2940291174943669}{3298534883328} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 18 a - 79\) , \( -84 a + 49\bigr] \) |
${y}^2+{x}{y}={x}^3-a{x}^2+\left(18a-79\right){x}-84a+49$ |
12.2-b4 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{20} \cdot 3^{10} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.473101729$ |
1.112282735 |
\( \frac{5735339}{3888} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -5 a + 16\) , \( 3 a - 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(-5a+16\right){x}+3a-9$ |
12.2-b5 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{16} \cdot 3^{20} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.736550864$ |
1.112282735 |
\( \frac{476379541}{236196} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 15 a - 44\) , \( 19 a + 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(15a-44\right){x}+19a+7$ |
12.2-b6 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{10} \cdot 3^{5} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.473101729$ |
1.112282735 |
\( \frac{572452561}{6912} a + \frac{201107101}{6912} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( -2 a + 6\) , \( 4\bigr] \) |
${y}^2+{x}{y}={x}^3-a{x}^2+\left(-2a+6\right){x}+4$ |
12.2-b7 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{10} \cdot 3^{17} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.473101729$ |
1.112282735 |
\( -\frac{572452561}{6912} a + \frac{386779831}{3456} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 19 a + 64\) , \( 54 a - 353\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(19a+64\right){x}+54a-353$ |
12.2-b8 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{32} \cdot 3^{4} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{3} \cdot 5 \) |
$1$ |
$0.347310172$ |
1.112282735 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 5460 a - 16379\) , \( -389123 a + 485305\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(5460a-16379\right){x}-389123a+485305$ |
22.2-a1 |
22.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{21} \cdot 11 \) |
$1.20858$ |
$(2,a), (11,a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$1$ |
$2.119611413$ |
0.678818924 |
\( \frac{1116919335759}{5632} a - \frac{145436402365}{5632} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -87 a + 184\) , \( a + 1820\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-87a+184\right){x}+a+1820$ |
22.2-a2 |
22.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{13} \cdot 11^{9} \) |
$1.20858$ |
$(2,a), (11,a+7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$2.119611413$ |
0.678818924 |
\( \frac{1607003454799}{4715895382} a - \frac{3364424472765}{4715895382} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 8 a - 21\) , \( 27 a - 50\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(8a-21\right){x}+27a-50$ |
22.2-a3 |
22.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11^{3} \) |
$1.20858$ |
$(2,a), (11,a+7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1$ |
$6.358834240$ |
0.678818924 |
\( -\frac{2138409}{10648} a + \frac{8227115}{10648} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -2 a + 9\) , \( -a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-2a+9\right){x}-a+2$ |
22.2-b1 |
22.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{21} \cdot 11 \) |
$1.20858$ |
$(2,a), (11,a+7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$2.119611413$ |
0.678818924 |
\( \frac{1116919335759}{5632} a - \frac{145436402365}{5632} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( 61 a + 202\) , \( 350 a - 1813\bigr] \) |
${y}^2+a{y}={x}^3+{x}^2+\left(61a+202\right){x}+350a-1813$ |
22.2-b2 |
22.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{13} \cdot 11^{9} \) |
$1.20858$ |
$(2,a), (11,a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$1$ |
$2.119611413$ |
0.678818924 |
\( \frac{1607003454799}{4715895382} a - \frac{3364424472765}{4715895382} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( -9 a - 18\) , \( -42 a + 27\bigr] \) |
${y}^2+a{y}={x}^3+{x}^2+\left(-9a-18\right){x}-42a+27$ |
22.2-b3 |
22.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11^{3} \) |
$1.20858$ |
$(2,a), (11,a+7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1$ |
$6.358834240$ |
0.678818924 |
\( -\frac{2138409}{10648} a + \frac{8227115}{10648} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( a + 2\) , \( a + 1\bigr] \) |
${y}^2+a{y}={x}^3+{x}^2+\left(a+2\right){x}+a+1$ |
22.3-a1 |
22.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
22.3 |
\( 2 \cdot 11 \) |
\( 2^{13} \cdot 11^{9} \) |
$1.20858$ |
$(2,a+1), (11,a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$2.119611413$ |
0.678818924 |
\( -\frac{1607003454799}{4715895382} a - \frac{878710508983}{2357947691} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -10 a - 11\) , \( -28 a - 22\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-10a-11\right){x}-28a-22$ |
22.3-a2 |
22.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
22.3 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11^{3} \) |
$1.20858$ |
$(2,a+1), (11,a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1$ |
$6.358834240$ |
0.678818924 |
\( \frac{2138409}{10648} a + \frac{3044353}{5324} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 9\) , \( 2\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+9{x}+2$ |
22.3-a3 |
22.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
22.3 |
\( 2 \cdot 11 \) |
\( 2^{21} \cdot 11 \) |
$1.20858$ |
$(2,a+1), (11,a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$1$ |
$2.119611413$ |
0.678818924 |
\( -\frac{1116919335759}{5632} a + \frac{485741466697}{2816} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 85 a + 99\) , \( -2 a + 1822\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(85a+99\right){x}-2a+1822$ |
22.3-b1 |
22.3-b |
$3$ |
$9$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
22.3 |
\( 2 \cdot 11 \) |
\( 2^{13} \cdot 11^{9} \) |
$1.20858$ |
$(2,a+1), (11,a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$1$ |
$2.119611413$ |
0.678818924 |
\( -\frac{1607003454799}{4715895382} a - \frac{878710508983}{2357947691} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( 9 a - 27\) , \( 41 a - 15\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+{x}^2+\left(9a-27\right){x}+41a-15$ |
22.3-b2 |
22.3-b |
$3$ |
$9$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
22.3 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11^{3} \) |
$1.20858$ |
$(2,a+1), (11,a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1$ |
$6.358834240$ |
0.678818924 |
\( \frac{2138409}{10648} a + \frac{3044353}{5324} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( -a + 3\) , \( -2 a + 2\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+{x}^2+\left(-a+3\right){x}-2a+2$ |
22.3-b3 |
22.3-b |
$3$ |
$9$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
22.3 |
\( 2 \cdot 11 \) |
\( 2^{21} \cdot 11 \) |
$1.20858$ |
$(2,a+1), (11,a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$2.119611413$ |
0.678818924 |
\( -\frac{1116919335759}{5632} a + \frac{485741466697}{2816} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( -61 a + 263\) , \( -351 a - 1463\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+{x}^2+\left(-61a+263\right){x}-351a-1463$ |
39.1-a1 |
39.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 2^{12} \cdot 3 \cdot 13 \) |
$1.39456$ |
$(3,a+1), (13,a+6)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1.432523755$ |
$7.561180171$ |
0.867219670 |
\( \frac{1359070}{39} a - \frac{4805891}{39} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(a-1\right){x}^2+{x}$ |
39.1-a2 |
39.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 2^{12} \cdot 3 \cdot 13 \) |
$1.39456$ |
$(3,a+1), (13,a+6)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1.432523755$ |
$7.561180171$ |
0.867219670 |
\( -\frac{1359070}{39} a - \frac{3446821}{39} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -4 a - 4\) , \( a + 25\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+a{x}^2+\left(-4a-4\right){x}+a+25$ |
39.1-a3 |
39.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{14} \cdot 13^{2} \) |
$1.39456$ |
$(3,a+1), (13,a+6)$ |
$2$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.432523755$ |
$7.561180171$ |
0.867219670 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+4{x}+6$ |
39.1-a4 |
39.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{16} \cdot 13^{4} \) |
$1.39456$ |
$(3,a+1), (13,a+6)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.358130938$ |
$3.780590085$ |
0.867219670 |
\( \frac{10218313}{1521} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -41\) , \( 96\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-41{x}+96$ |
39.1-a5 |
39.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{14} \cdot 13^{8} \) |
$1.39456$ |
$(3,a+1), (13,a+6)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.089532734$ |
$1.890295042$ |
0.867219670 |
\( \frac{822656953}{85683} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -176\) , \( -768\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-176{x}-768$ |
39.1-a6 |
39.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{20} \cdot 13^{2} \) |
$1.39456$ |
$(3,a+1), (13,a+6)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.358130938$ |
$1.890295042$ |
0.867219670 |
\( \frac{37159393753}{1053} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -626\) , \( 6180\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-626{x}+6180$ |
39.1-b1 |
39.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 2^{12} \cdot 3 \cdot 13 \) |
$1.39456$ |
$(3,a+1), (13,a+6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 1 \) |
$1$ |
$7.561180171$ |
2.421515642 |
\( \frac{1359070}{39} a - \frac{4805891}{39} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -3 a + 9\) , \( 2 a - 10\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3-{x}^2+\left(-3a+9\right){x}+2a-10$ |
39.1-b2 |
39.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 2^{12} \cdot 3 \cdot 13 \) |
$1.39456$ |
$(3,a+1), (13,a+6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 1 \) |
$1$ |
$7.561180171$ |
2.421515642 |
\( -\frac{1359070}{39} a - \frac{3446821}{39} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( 3 a + 6\) , \( -2 a - 8\bigr] \) |
${y}^2+a{x}{y}={x}^3-a{x}^2+\left(3a+6\right){x}-2a-8$ |
39.1-b3 |
39.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{2} \cdot 13^{2} \) |
$1.39456$ |
$(3,a+1), (13,a+6)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$7.561180171$ |
2.421515642 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2+{x}$ |
39.1-b4 |
39.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{4} \cdot 13^{4} \) |
$1.39456$ |
$(3,a+1), (13,a+6)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$3.780590085$ |
2.421515642 |
\( \frac{10218313}{1521} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -4\) , \( -5\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2-4{x}-5$ |
*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.