| 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 |
| 192.4-a1 |
192.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{17} \cdot 3^{14} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \cdot 7 \) |
$0.094252160$ |
$6.770407563$ |
5.916525472 |
\( -\frac{1638090289}{69984} a + \frac{3499732859}{34992} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 98 a - 413\) , \( -926 a + 3963\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(98a-413\right){x}-926a+3963$ |
| 192.4-a2 |
192.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{16} \cdot 3^{7} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \cdot 7 \) |
$0.047126080$ |
$13.54081512$ |
5.916525472 |
\( \frac{2229520813}{82944} a + \frac{3649348529}{41472} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 3 a - 8\) , \( -31 a + 134\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(3a-8\right){x}-31a+134$ |
| 192.4-b1 |
192.4-b |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{13} \cdot 3^{2} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.147835330$ |
$7.492300502$ |
2.278176375 |
\( -\frac{38131841}{6} a + 27168441 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 17672 a - 75536\) , \( 2506014 a - 10712996\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(17672a-75536\right){x}+2506014a-10712996$ |
| 192.4-b2 |
192.4-b |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.573917665$ |
$14.98460100$ |
2.278176375 |
\( \frac{40807}{12} a + \frac{29555}{6} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 1067 a - 4551\) , \( 42938 a - 183550\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(1067a-4551\right){x}+42938a-183550$ |
| 192.4-c1 |
192.4-c |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( 2^{14} \cdot 3^{14} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 7 \) |
$0.055980386$ |
$7.352463743$ |
3.052947854 |
\( -\frac{24817}{324} a + \frac{9059129}{4374} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -37957 a - 124309\) , \( 684104 a + 2240382\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-37957a-124309\right){x}+684104a+2240382$ |
| 192.4-c2 |
192.4-c |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{10} \cdot 3^{7} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$0.111960773$ |
$14.70492748$ |
3.052947854 |
\( \frac{67650661}{1296} a + \frac{111597257}{648} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -27372 a - 89644\) , \( 4737215 a + 15513985\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-27372a-89644\right){x}+4737215a+15513985$ |
| 192.4-d1 |
192.4-d |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{25} \cdot 3^{2} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$9.664863977$ |
5.120570026 |
\( -\frac{75000761}{196608} a + \frac{51906161}{32768} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -a - 2\) , \( 13 a + 42\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-2\right){x}+13a+42$ |
| 192.4-d2 |
192.4-d |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$2.416215994$ |
5.120570026 |
\( -\frac{98620637623651}{48} a + \frac{70265843645899}{8} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -15841 a - 51878\) , \( -273555 a - 895870\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-15841a-51878\right){x}-273555a-895870$ |
| 192.4-d3 |
192.4-d |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( 2^{14} \cdot 3^{4} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$9.664863977$ |
5.120570026 |
\( -\frac{1139920067}{2304} a + \frac{2671449217}{1152} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -10276 a - 33653\) , \( 1063953 a + 3484358\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-10276a-33653\right){x}+1063953a+3484358$ |
| 192.4-d4 |
192.4-d |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{13} \cdot 3^{8} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$4.832431988$ |
5.120570026 |
\( \frac{7086201848561}{1296} a + \frac{1289262372893}{72} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -164286 a - 538023\) , \( 69887187 a + 228874752\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-164286a-538023\right){x}+69887187a+228874752$ |
| 192.4-e1 |
192.4-e |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{22} \cdot 3^{3} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.205284479$ |
$11.74665338$ |
5.110375634 |
\( \frac{2256186877}{589824} a - \frac{3477920191}{294912} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -4112 a - 13460\) , \( -219994 a - 720457\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-4112a-13460\right){x}-219994a-720457$ |
| 192.4-e2 |
192.4-e |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( 2^{20} \cdot 3^{6} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.102642239$ |
$11.74665338$ |
5.110375634 |
\( -\frac{574770342811}{6912} a + \frac{409533665347}{1152} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -21317 a - 69805\) , \( 3098729 a + 10148086\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-21317a-69805\right){x}+3098729a+10148086$ |
| 192.4-f1 |
192.4-f |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{22} \cdot 3^{3} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$4.773717125$ |
1.896882838 |
\( \frac{2256186877}{589824} a - \frac{3477920191}{294912} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 43 a - 161\) , \( 279 a - 1163\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(43a-161\right){x}+279a-1163$ |
| 192.4-f2 |
192.4-f |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( 2^{20} \cdot 3^{6} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$2.386858562$ |
1.896882838 |
\( -\frac{574770342811}{6912} a + \frac{409533665347}{1152} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 638 a - 2706\) , \( 16991 a - 72611\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(638a-2706\right){x}+16991a-72611$ |
| 192.4-g1 |
192.4-g |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{25} \cdot 3^{2} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.912877992$ |
$6.004569731$ |
2.904137626 |
\( -\frac{75000761}{196608} a + \frac{51906161}{32768} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( 13546 a - 57908\) , \( -1014720 a + 4337844\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(13546a-57908\right){x}-1014720a+4337844$ |
| 192.4-g2 |
192.4-g |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.912877992$ |
$12.00913946$ |
2.904137626 |
\( -\frac{98620637623651}{48} a + \frac{70265843645899}{8} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( 1106 a - 4718\) , \( -39383 a + 168369\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1106a-4718\right){x}-39383a+168369$ |
| 192.4-g3 |
192.4-g |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( 2^{14} \cdot 3^{4} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.825755985$ |
$12.00913946$ |
2.904137626 |
\( -\frac{1139920067}{2304} a + \frac{2671449217}{1152} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( 71 a - 293\) , \( -641 a + 2751\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(71a-293\right){x}-641a+2751$ |
| 192.4-g4 |
192.4-g |
$4$ |
$4$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{13} \cdot 3^{8} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.651511970$ |
$3.002284865$ |
2.904137626 |
\( \frac{7086201848561}{1296} a + \frac{1289262372893}{72} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( 61 a - 263\) , \( -795 a + 3357\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(61a-263\right){x}-795a+3357$ |
| 192.4-h1 |
192.4-h |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( 2^{14} \cdot 3^{14} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.201234166$ |
1.642747061 |
\( -\frac{24817}{324} a + \frac{9059129}{4374} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 10 a - 55\) , \( 47 a - 206\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(10a-55\right){x}+47a-206$ |
| 192.4-h2 |
192.4-h |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{10} \cdot 3^{7} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.201234166$ |
1.642747061 |
\( \frac{67650661}{1296} a + \frac{111597257}{648} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -5 a + 10\) , \( -4 a + 11\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-5a+10\right){x}-4a+11$ |
| 192.4-i1 |
192.4-i |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{13} \cdot 3^{2} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$13.47105648$ |
3.568570040 |
\( -\frac{38131841}{6} a + 27168441 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( a + 10\) , \( -21 a - 66\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+10\right){x}-21a-66$ |
| 192.4-i2 |
192.4-i |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$26.94211297$ |
3.568570040 |
\( \frac{40807}{12} a + \frac{29555}{6} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -4 a - 5\) , \( -5 a - 12\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4a-5\right){x}-5a-12$ |
| 192.4-j1 |
192.4-j |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{17} \cdot 3^{14} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.811458348$ |
0.479867039 |
\( -\frac{1638090289}{69984} a + \frac{3499732859}{34992} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -12271 a - 40182\) , \( -8930495 a - 29246630\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-12271a-40182\right){x}-8930495a-29246630$ |
| 192.4-j2 |
192.4-j |
$2$ |
$2$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
192.4 |
\( 2^{6} \cdot 3 \) |
\( - 2^{16} \cdot 3^{7} \) |
$2.51132$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.622916697$ |
0.479867039 |
\( \frac{2229520813}{82944} a + \frac{3649348529}{41472} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -25766 a - 84377\) , \( -4403445 a - 14420916\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-25766a-84377\right){x}-4403445a-14420916$ |
*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.
