| 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 |
| 16.1-a1 |
16.1-a |
$6$ |
$18$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{18} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$0.145659266$ |
1.125565163 |
\( -\frac{21417711216510281745097895}{64} a^{3} - \frac{370966176885404047689177}{8} a^{2} + \frac{6481850338880742243324709}{4} a + 1175697316394585477461175 \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 3\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{3}{2} a + 4\) , \( a^{2} - a - 2\) , \( -63 a^{3} - 511 a^{2} - 334 a - 99\) , \( -\frac{7331}{2} a^{3} - \frac{23441}{2} a^{2} - \frac{5689}{2} a + 2767\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-3\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{3}{2}a+4\right){x}^{2}+\left(-63a^{3}-511a^{2}-334a-99\right){x}-\frac{7331}{2}a^{3}-\frac{23441}{2}a^{2}-\frac{5689}{2}a+2767$ |
| 16.1-a2 |
16.1-a |
$6$ |
$18$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{54} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{4} \) |
$1$ |
$11.79840057$ |
1.125565163 |
\( -\frac{4805055317535}{262144} a^{3} - \frac{83328303025}{32768} a^{2} + \frac{1454285809955}{16384} a + \frac{2110395388989}{32768} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 3\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{3}{2} a + 4\) , \( a^{2} - a - 2\) , \( -3 a^{3} - 11 a^{2} + 6 a + 21\) , \( \frac{13}{2} a^{3} - \frac{17}{2} a^{2} - \frac{57}{2} a - 9\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-3\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{3}{2}a+4\right){x}^{2}+\left(-3a^{3}-11a^{2}+6a+21\right){x}+\frac{13}{2}a^{3}-\frac{17}{2}a^{2}-\frac{57}{2}a-9$ |
| 16.1-a3 |
16.1-a |
$6$ |
$18$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{18} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/18\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$955.6704461$ |
1.125565163 |
\( \frac{546880529}{128} a^{3} - \frac{995601459}{128} a^{2} - \frac{1917966475}{128} a + \frac{666851395}{32} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{3}{2} a - 3\) , \( a - 1\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 2\) , \( -a^{3} + 2 a^{2} + 4 a - 4\) , \( a^{3} - 2 a^{2} - 3 a + 7\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{3}{2}a-3\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-2\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a^{3}+2a^{2}+4a-4\right){x}+a^{3}-2a^{2}-3a+7$ |
| 16.1-a4 |
16.1-a |
$6$ |
$18$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{18} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/18\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$955.6704461$ |
1.125565163 |
\( \frac{45964405}{32} a^{3} + \frac{132431655}{64} a^{2} - \frac{68561993}{32} a - \frac{37780611}{16} \) |
\( \bigl[a^{2} - 3\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + 1\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + 1\) , \( a^{2} - 2 a\) , \( \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{1}{2} a + 1\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+1\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+1\right){x}^{2}+\left(a^{2}-2a\right){x}+\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-\frac{1}{2}a+1$ |
| 16.1-a5 |
16.1-a |
$6$ |
$18$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{18} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$0.145659266$ |
1.125565163 |
\( -\frac{13882195646026523479452721}{64} a^{3} + \frac{1195863633675626175189501}{2} a^{2} + \frac{273764395689817472557063}{8} a - \frac{31611543767746583544251007}{64} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 2\) , \( \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{1}{2} a + 2\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a\) , \( -610 a^{3} + 1181 a^{2} + 2357 a - 4063\) , \( -\frac{39313}{2} a^{3} + \frac{73397}{2} a^{2} + \frac{143695}{2} a - 109064\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-2\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}a\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-\frac{1}{2}a+2\right){x}^{2}+\left(-610a^{3}+1181a^{2}+2357a-4063\right){x}-\frac{39313}{2}a^{3}+\frac{73397}{2}a^{2}+\frac{143695}{2}a-109064$ |
| 16.1-a6 |
16.1-a |
$6$ |
$18$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{54} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{4} \) |
$1$ |
$11.79840057$ |
1.125565163 |
\( -\frac{3114192685065}{262144} a^{3} + \frac{536617151675}{16384} a^{2} + \frac{61146381065}{32768} a - \frac{7093520331823}{262144} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 2\) , \( \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{1}{2} a + 2\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a\) , \( -10 a^{3} + 21 a^{2} + 37 a - 63\) , \( -\frac{33}{2} a^{3} + \frac{69}{2} a^{2} + \frac{111}{2} a - 104\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-2\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}a\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-\frac{1}{2}a+2\right){x}^{2}+\left(-10a^{3}+21a^{2}+37a-63\right){x}-\frac{33}{2}a^{3}+\frac{69}{2}a^{2}+\frac{111}{2}a-104$ |
| 16.1-b1 |
16.1-b |
$8$ |
$20$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{50} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.4[2] |
$25$ |
\( 2 \) |
$1$ |
$2.751400638$ |
0.729119711 |
\( \frac{888291634749271321}{2097152} a^{3} - \frac{1616912130457070213}{2097152} a^{2} - \frac{3115187239876266733}{2097152} a + \frac{2165906537306684017}{1048576} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 3\) , \( 1\) , \( 0\) , \( -\frac{91}{2} a^{3} - \frac{131}{2} a^{2} + \frac{145}{2} a + 64\) , \( -\frac{1923}{2} a^{3} - \frac{2783}{2} a^{2} + \frac{2849}{2} a + 1546\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-3\right){x}{y}={x}^{3}+{x}^{2}+\left(-\frac{91}{2}a^{3}-\frac{131}{2}a^{2}+\frac{145}{2}a+64\right){x}-\frac{1923}{2}a^{3}-\frac{2783}{2}a^{2}+\frac{2849}{2}a+1546$ |
| 16.1-b2 |
16.1-b |
$8$ |
$20$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{50} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.4[2] |
$25$ |
\( 2 \) |
$1$ |
$2.751400638$ |
0.729119711 |
\( \frac{149412609540572745}{1048576} a^{3} + \frac{214897638313326701}{1048576} a^{2} - \frac{111540830403745925}{524288} a - \frac{61277779407734761}{262144} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{3}{2} a - 3\) , \( -a^{2} + 2 a + 3\) , \( a^{2} - a - 3\) , \( 7 a^{3} + 11 a^{2} - 12 a - 132\) , \( -\frac{31}{2} a^{3} + \frac{449}{2} a^{2} + \frac{27}{2} a - 1143\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{3}{2}a-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(7a^{3}+11a^{2}-12a-132\right){x}-\frac{31}{2}a^{3}+\frac{449}{2}a^{2}+\frac{27}{2}a-1143$ |
| 16.1-b3 |
16.1-b |
$8$ |
$20$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{20} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.4[2] |
$100$ |
\( 1 \) |
$1$ |
$1.375700319$ |
0.729119711 |
\( -\frac{385603532444261137105}{64} a^{3} + \frac{385603532444261137105}{64} a^{2} + \frac{1156810597332783411315}{64} a + \frac{238316089232576270321}{32} \) |
\( \bigl[a^{2} - 3\) , \( -\frac{1}{2} a^{3} + \frac{3}{2} a^{2} + \frac{1}{2} a - 3\) , \( a^{2} - 3\) , \( \frac{1373}{2} a^{3} - \frac{4061}{2} a^{2} + \frac{341}{2} a + 1401\) , \( \frac{57843}{2} a^{3} - \frac{160487}{2} a^{2} - \frac{7077}{2} a + 65326\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{3}{2}a^{2}+\frac{1}{2}a-3\right){x}^{2}+\left(\frac{1373}{2}a^{3}-\frac{4061}{2}a^{2}+\frac{341}{2}a+1401\right){x}+\frac{57843}{2}a^{3}-\frac{160487}{2}a^{2}-\frac{7077}{2}a+65326$ |
| 16.1-b4 |
16.1-b |
$8$ |
$20$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{10} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1[2] |
$1$ |
\( 2 \) |
$1$ |
$1719.625399$ |
0.729119711 |
\( -\frac{50109851}{8} a^{3} - \frac{13881241}{16} a^{2} + \frac{242645051}{8} a + \frac{88020739}{4} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 2\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 2\) , \( -a^{3} - 4 a^{2} - 3 a + 2\) , \( \frac{7}{2} a^{3} + \frac{9}{2} a^{2} - \frac{11}{2} a - 5\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-2\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-a^{3}-4a^{2}-3a+2\right){x}+\frac{7}{2}a^{3}+\frac{9}{2}a^{2}-\frac{11}{2}a-5$ |
| 16.1-b5 |
16.1-b |
$8$ |
$20$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1[2] |
$4$ |
\( 1 \) |
$1$ |
$859.8126995$ |
0.729119711 |
\( \frac{908026183625}{4} a^{3} - \frac{908026183625}{4} a^{2} - \frac{2724078550875}{4} a + 734608622353 \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a\) , \( -\frac{1}{2} a^{3} + \frac{3}{2} a^{2} - \frac{1}{2} a - 2\) , \( a^{2} - 3\) , \( -a^{3} - 9 a^{2} + 22 a - 12\) , \( \frac{141}{2} a^{3} - \frac{347}{2} a^{2} - \frac{121}{2} a + 189\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{3}{2}a^{2}-\frac{1}{2}a-2\right){x}^{2}+\left(-a^{3}-9a^{2}+22a-12\right){x}+\frac{141}{2}a^{3}-\frac{347}{2}a^{2}-\frac{121}{2}a+189$ |
| 16.1-b6 |
16.1-b |
$8$ |
$20$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{40} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2Cs, 5B.1.4[2] |
$25$ |
\( 2^{2} \) |
$1$ |
$5.502801277$ |
0.729119711 |
\( -\frac{2376401567779}{2048} a^{3} + \frac{2376401567779}{2048} a^{2} + \frac{7129204703337}{2048} a + \frac{735163478721}{512} \) |
\( \bigl[a + 1\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 4\) , \( a^{2} - 2\) , \( 507 a^{3} + 75 a^{2} - 2471 a - 1833\) , \( 11504 a^{3} + 1595 a^{2} - 55760 a - 40525\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-4\right){x}^{2}+\left(507a^{3}+75a^{2}-2471a-1833\right){x}+11504a^{3}+1595a^{2}-55760a-40525$ |
| 16.1-b7 |
16.1-b |
$8$ |
$20$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2Cs, 5B.1.1[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$3439.250798$ |
0.729119711 |
\( \frac{1790195}{8} a^{3} - \frac{1790195}{8} a^{2} - \frac{5370585}{8} a + \frac{2926869}{4} \) |
\( \bigl[a + 1\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 4\) , \( a^{2} - 2\) , \( 2 a^{3} - 11 a - 8\) , \( -10 a^{3} - 2 a^{2} + 48 a + 36\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-4\right){x}^{2}+\left(2a^{3}-11a-8\right){x}-10a^{3}-2a^{2}+48a+36$ |
| 16.1-b8 |
16.1-b |
$8$ |
$20$ |
4.4.2225.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{10} \) |
$5.96099$ |
$(-a), (-1/2a^3+1/2a^2+5/2a-1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1[2] |
$1$ |
\( 2 \) |
$1$ |
$1719.625399$ |
0.729119711 |
\( -\frac{129909351}{32} a^{3} + \frac{358111237}{32} a^{2} + \frac{20466061}{32} a - \frac{73945853}{8} \) |
\( \bigl[a + 1\) , \( a^{2} - 2 a - 3\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 3\) , \( a^{3} - 7 a^{2} + 8 a + 6\) , \( -3 a^{3} + 11 a^{2} - 5 a - 10\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-3\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(a^{3}-7a^{2}+8a+6\right){x}-3a^{3}+11a^{2}-5a-10$ |
*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.
