| 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 |
| 19.1-a1 |
19.1-a |
$2$ |
$3$ |
4.4.16400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{12} \) |
$16.53502$ |
$(a^2+a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.403755163$ |
$5.505102201$ |
2.896516895 |
\( -\frac{260081870579892941420531712}{2213314919066161} a^{3} - \frac{717846408960670827499892736}{2213314919066161} a^{2} + \frac{1399751787087143777899143168}{2213314919066161} a + \frac{3863424973120559478115966976}{2213314919066161} \) |
\( \bigl[0\) , \( a^{2} - 8\) , \( a^{3} + a^{2} - 6 a - 7\) , \( -189 a^{3} + 430 a^{2} + 1431 a - 3272\) , \( -77400 a^{3} + 179546 a^{2} + 589627 a - 1367803\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-7\right){y}={x}^{3}+\left(a^{2}-8\right){x}^{2}+\left(-189a^{3}+430a^{2}+1431a-3272\right){x}-77400a^{3}+179546a^{2}+589627a-1367803$ |
| 19.1-a2 |
19.1-a |
$2$ |
$3$ |
4.4.16400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{4} \) |
$16.53502$ |
$(a^2+a-4)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.467918387$ |
$445.9132782$ |
2.896516895 |
\( \frac{64098067009536}{130321} a^{3} + \frac{148700151410688}{130321} a^{2} - \frac{488300922445824}{130321} a - \frac{1132805100666880}{130321} \) |
\( \bigl[0\) , \( a^{2} - 8\) , \( a^{3} + a^{2} - 6 a - 7\) , \( 21 a^{3} - 50 a^{2} - 159 a + 378\) , \( 2850 a^{3} - 6613 a^{2} - 21713 a + 50376\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-7\right){y}={x}^{3}+\left(a^{2}-8\right){x}^{2}+\left(21a^{3}-50a^{2}-159a+378\right){x}+2850a^{3}-6613a^{2}-21713a+50376$ |
| 19.1-b1 |
19.1-b |
$4$ |
$15$ |
4.4.16400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{3} \) |
$16.53502$ |
$(a^2+a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B[2] |
$1$ |
\( 3 \) |
$5.459177605$ |
$6.388084289$ |
3.267812905 |
\( \frac{6630368958622669095}{6859} a^{3} - \frac{15410236864329038175}{6859} a^{2} - \frac{50366099200344265580}{6859} a + \frac{116997492663148095175}{6859} \) |
\( \bigl[a^{3} + a^{2} - 7 a - 7\) , \( -a^{3} - a^{2} + 8 a + 6\) , \( a^{3} - 6 a\) , \( -817 a^{3} - 2237 a^{2} + 4498 a + 12277\) , \( 37692 a^{3} + 105138 a^{2} - 197236 a - 552811\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-7a-7\right){x}{y}+\left(a^{3}-6a\right){y}={x}^{3}+\left(-a^{3}-a^{2}+8a+6\right){x}^{2}+\left(-817a^{3}-2237a^{2}+4498a+12277\right){x}+37692a^{3}+105138a^{2}-197236a-552811$ |
| 19.1-b2 |
19.1-b |
$4$ |
$15$ |
4.4.16400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{15} \) |
$16.53502$ |
$(a^2+a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B[2] |
$1$ |
\( 3 \cdot 5 \) |
$1.091835521$ |
$6.388084289$ |
3.267812905 |
\( \frac{35837154163366394281545}{15181127029874798299} a^{3} + \frac{65378672019532107253000}{15181127029874798299} a^{2} - \frac{272832802000467030921805}{15181127029874798299} a - \frac{500017660482708507945475}{15181127029874798299} \) |
\( \bigl[a^{3} - 6 a\) , \( a + 1\) , \( a^{3} + a^{2} - 7 a - 6\) , \( -50 a^{3} + 117 a^{2} + 413 a - 962\) , \( -92 a^{3} + 120 a^{2} + 1231 a - 2356\bigr] \) |
${y}^2+\left(a^{3}-6a\right){x}{y}+\left(a^{3}+a^{2}-7a-6\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-50a^{3}+117a^{2}+413a-962\right){x}-92a^{3}+120a^{2}+1231a-2356$ |
| 19.1-b3 |
19.1-b |
$4$ |
$15$ |
4.4.16400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{5} \) |
$16.53502$ |
$(a^2+a-4)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B[2] |
$1$ |
\( 5 \) |
$0.363945173$ |
$517.4348274$ |
3.267812905 |
\( \frac{2289565641755}{2476099} a^{3} - \frac{5313278941225}{2476099} a^{2} - \frac{17442014714520}{2476099} a + \frac{40477042159125}{2476099} \) |
\( \bigl[a^{3} - 6 a\) , \( a + 1\) , \( a^{3} + a^{2} - 7 a - 6\) , \( -30 a^{3} + 77 a^{2} + 228 a - 567\) , \( 198 a^{3} - 445 a^{2} - 1525 a + 3458\bigr] \) |
${y}^2+\left(a^{3}-6a\right){x}{y}+\left(a^{3}+a^{2}-7a-6\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-30a^{3}+77a^{2}+228a-567\right){x}+198a^{3}-445a^{2}-1525a+3458$ |
| 19.1-b4 |
19.1-b |
$4$ |
$15$ |
4.4.16400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( -19 \) |
$16.53502$ |
$(a^2+a-4)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B[2] |
$1$ |
\( 1 \) |
$1.819725868$ |
$517.4348274$ |
3.267812905 |
\( -\frac{24005877695}{19} a^{3} - \frac{66262595100}{19} a^{2} + \frac{129187416855}{19} a + \frac{356600683675}{19} \) |
\( \bigl[a^{3} - 6 a\) , \( -a^{2} - a + 6\) , \( a^{2} - 6\) , \( -12 a^{3} + 24 a^{2} + 102 a - 216\) , \( -73 a^{3} + 154 a^{2} + 633 a - 1386\bigr] \) |
${y}^2+\left(a^{3}-6a\right){x}{y}+\left(a^{2}-6\right){y}={x}^{3}+\left(-a^{2}-a+6\right){x}^{2}+\left(-12a^{3}+24a^{2}+102a-216\right){x}-73a^{3}+154a^{2}+633a-1386$ |
| 19.1-c1 |
19.1-c |
$4$ |
$15$ |
4.4.16400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{3} \) |
$16.53502$ |
$(a^2+a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B[2] |
$1$ |
\( 1 \) |
$1$ |
$98.99387271$ |
0.773012275 |
\( \frac{6630368958622669095}{6859} a^{3} - \frac{15410236864329038175}{6859} a^{2} - \frac{50366099200344265580}{6859} a + \frac{116997492663148095175}{6859} \) |
\( \bigl[a^{2} + a - 6\) , \( a^{3} - a^{2} - 6 a + 7\) , \( 0\) , \( -809 a^{3} - 2216 a^{2} + 4441 a + 12136\) , \( -40319 a^{3} - 112303 a^{2} + 211833 a + 592438\bigr] \) |
${y}^2+\left(a^{2}+a-6\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-6a+7\right){x}^{2}+\left(-809a^{3}-2216a^{2}+4441a+12136\right){x}-40319a^{3}-112303a^{2}+211833a+592438$ |
| 19.1-c2 |
19.1-c |
$4$ |
$15$ |
4.4.16400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{15} \) |
$16.53502$ |
$(a^2+a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B[2] |
$1$ |
\( 1 \) |
$1$ |
$98.99387271$ |
0.773012275 |
\( \frac{35837154163366394281545}{15181127029874798299} a^{3} + \frac{65378672019532107253000}{15181127029874798299} a^{2} - \frac{272832802000467030921805}{15181127029874798299} a - \frac{500017660482708507945475}{15181127029874798299} \) |
\( \bigl[1\) , \( a^{2} - a - 6\) , \( a^{3} - 7 a + 1\) , \( -52 a^{3} + 113 a^{2} + 424 a - 939\) , \( 103 a^{3} - 150 a^{2} - 1235 a + 2384\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-7a+1\right){y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(-52a^{3}+113a^{2}+424a-939\right){x}+103a^{3}-150a^{2}-1235a+2384$ |
| 19.1-c3 |
19.1-c |
$4$ |
$15$ |
4.4.16400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{5} \) |
$16.53502$ |
$(a^2+a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B[2] |
$1$ |
\( 1 \) |
$1$ |
$98.99387271$ |
0.773012275 |
\( \frac{2289565641755}{2476099} a^{3} - \frac{5313278941225}{2476099} a^{2} - \frac{17442014714520}{2476099} a + \frac{40477042159125}{2476099} \) |
\( \bigl[1\) , \( a^{2} - a - 6\) , \( a^{3} - 7 a + 1\) , \( -32 a^{3} + 73 a^{2} + 239 a - 544\) , \( -212 a^{3} + 490 a^{2} + 1626 a - 3765\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-7a+1\right){y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(-32a^{3}+73a^{2}+239a-544\right){x}-212a^{3}+490a^{2}+1626a-3765$ |
| 19.1-c4 |
19.1-c |
$4$ |
$15$ |
4.4.16400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( -19 \) |
$16.53502$ |
$(a^2+a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B[2] |
$1$ |
\( 1 \) |
$1$ |
$98.99387271$ |
0.773012275 |
\( -\frac{24005877695}{19} a^{3} - \frac{66262595100}{19} a^{2} + \frac{129187416855}{19} a + \frac{356600683675}{19} \) |
\( \bigl[1\) , \( -a^{2} + a + 7\) , \( a^{3} - 7 a + 1\) , \( -12 a^{3} + 22 a^{2} + 105 a - 205\) , \( 61 a^{3} - 130 a^{2} - 529 a + 1166\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-7a+1\right){y}={x}^{3}+\left(-a^{2}+a+7\right){x}^{2}+\left(-12a^{3}+22a^{2}+105a-205\right){x}+61a^{3}-130a^{2}-529a+1166$ |
| 19.1-d1 |
19.1-d |
$2$ |
$3$ |
4.4.16400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{12} \) |
$16.53502$ |
$(a^2+a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$34.85936199$ |
0.544411769 |
\( -\frac{260081870579892941420531712}{2213314919066161} a^{3} - \frac{717846408960670827499892736}{2213314919066161} a^{2} + \frac{1399751787087143777899143168}{2213314919066161} a + \frac{3863424973120559478115966976}{2213314919066161} \) |
\( \bigl[0\) , \( -a^{2} + 8\) , \( a + 1\) , \( -189 a^{3} + 430 a^{2} + 1431 a - 3272\) , \( 77400 a^{3} - 179548 a^{2} - 589628 a + 1367811\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+8\right){x}^{2}+\left(-189a^{3}+430a^{2}+1431a-3272\right){x}+77400a^{3}-179548a^{2}-589628a+1367811$ |
| 19.1-d2 |
19.1-d |
$2$ |
$3$ |
4.4.16400.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{4} \) |
$16.53502$ |
$(a^2+a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$34.85936199$ |
0.544411769 |
\( \frac{64098067009536}{130321} a^{3} + \frac{148700151410688}{130321} a^{2} - \frac{488300922445824}{130321} a - \frac{1132805100666880}{130321} \) |
\( \bigl[0\) , \( -a^{2} + 8\) , \( a + 1\) , \( 21 a^{3} - 50 a^{2} - 159 a + 378\) , \( -2850 a^{3} + 6611 a^{2} + 21712 a - 50368\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+8\right){x}^{2}+\left(21a^{3}-50a^{2}-159a+378\right){x}-2850a^{3}+6611a^{2}+21712a-50368$ |
*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.
