| 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 |
| 171.1-a1 |
171.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{13} \cdot 19 \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1.660744489$ |
$3.604998134$ |
2.747015182 |
\( -\frac{12364905437067631}{124659} a - \frac{2836704000281954}{6561} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -659572 a - 2875005\) , \( 608482693 a + 2652314563\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-659572a-2875005\right){x}+608482693a+2652314563$ |
| 171.1-b1 |
171.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{4} \cdot 19^{10} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 2^{3} \) |
$1$ |
$0.085998375$ |
1.972938037 |
\( -\frac{9358714467168256}{22284891} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -4390\) , \( -113432\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-4390{x}-113432$ |
| 171.1-b2 |
171.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{20} \cdot 19^{2} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2^{3} \cdot 5^{2} \) |
$1$ |
$2.149959381$ |
1.972938037 |
\( \frac{841232384}{1121931} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 20\) , \( -32\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+20{x}-32$ |
| 171.1-c1 |
171.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{13} \cdot 19 \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1.660744489$ |
$3.604998134$ |
2.747015182 |
\( \frac{12364905437067631}{124659} a - \frac{2836704000281954}{6561} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( 659571 a - 2875005\) , \( -608482693 a + 2652314563\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(659571a-2875005\right){x}-608482693a+2652314563$ |
| 171.1-d1 |
171.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{13} \cdot 19 \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \cdot 5 \) |
$1$ |
$1.316496682$ |
6.040501053 |
\( -\frac{12364905437067631}{124659} a - \frac{2836704000281954}{6561} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( -659572 a - 2874998\) , \( -609801836 a - 2658064575\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-659572a-2874998\right){x}-609801836a-2658064575$ |
| 171.1-e1 |
171.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{4} \cdot 19^{2} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.014286018$ |
$30.86363048$ |
1.618457864 |
\( -\frac{1404928}{171} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -2\) , \( 2\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-2{x}+2$ |
| 171.1-f1 |
171.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( - 3^{10} \cdot 19 \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$5.203787826$ |
$0.565538563$ |
0.675157357 |
\( -\frac{80836867580206264835}{124659} a + \frac{18545249299658286056}{6561} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 175 a - 860\) , \( 3178 a - 14456\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(175a-860\right){x}+3178a-14456$ |
| 171.1-f2 |
171.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{2} \cdot 19^{8} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.601893913$ |
$2.262154252$ |
0.675157357 |
\( \frac{67419143}{390963} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 10\) , \( -10\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+10{x}-10$ |
| 171.1-f3 |
171.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{2} \cdot 19^{2} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.650473478$ |
$36.19446804$ |
0.675157357 |
\( \frac{389017}{57} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 0\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}$ |
| 171.1-f4 |
171.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{4} \cdot 19^{4} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.300946956$ |
$9.048617011$ |
0.675157357 |
\( \frac{30664297}{3249} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -5\) , \( -16\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-5{x}-16$ |
| 171.1-f5 |
171.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{8} \cdot 19^{2} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.601893913$ |
$2.262154252$ |
0.675157357 |
\( \frac{115714886617}{1539} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -100\) , \( -586\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-100{x}-586$ |
| 171.1-f6 |
171.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( - 3^{10} \cdot 19 \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$5.203787826$ |
$0.565538563$ |
0.675157357 |
\( \frac{80836867580206264835}{124659} a + \frac{18545249299658286056}{6561} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -175 a - 860\) , \( -3178 a - 14456\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-175a-860\right){x}-3178a-14456$ |
| 171.1-g1 |
171.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{13} \cdot 19 \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \cdot 5 \) |
$1$ |
$1.316496682$ |
6.040501053 |
\( \frac{12364905437067631}{124659} a - \frac{2836704000281954}{6561} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( 659571 a - 2874998\) , \( 609801836 a - 2658064575\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(659571a-2874998\right){x}+609801836a-2658064575$ |
| 171.1-h1 |
171.1-h |
$2$ |
$5$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{4} \cdot 19^{10} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.226284916$ |
$3.435258684$ |
7.133427351 |
\( -\frac{9358714467168256}{22284891} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -4390\) , \( 113427\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}-4390{x}+113427$ |
| 171.1-h2 |
171.1-h |
$2$ |
$5$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{20} \cdot 19^{2} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2^{3} \) |
$1.131424581$ |
$3.435258684$ |
7.133427351 |
\( \frac{841232384}{1121931} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( 20\) , \( 27\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+20{x}+27$ |
| 171.1-i1 |
171.1-i |
$6$ |
$8$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( - 3^{10} \cdot 19 \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$9.423048177$ |
1.080897756 |
\( -\frac{80836867580206264835}{124659} a + \frac{18545249299658286056}{6561} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 175 a - 862\) , \( -2828 a + 12735\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(175a-862\right){x}-2828a+12735$ |
| 171.1-i2 |
171.1-i |
$6$ |
$8$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{2} \cdot 19^{8} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$4.711524088$ |
1.080897756 |
\( \frac{67419143}{390963} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 8\) , \( 29\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+8{x}+29$ |
| 171.1-i3 |
171.1-i |
$6$ |
$8$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{2} \cdot 19^{2} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$18.84609635$ |
1.080897756 |
\( \frac{389017}{57} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-2{x}-1$ |
| 171.1-i4 |
171.1-i |
$6$ |
$8$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{4} \cdot 19^{4} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$18.84609635$ |
1.080897756 |
\( \frac{30664297}{3249} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -7\) , \( 5\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-7{x}+5$ |
| 171.1-i5 |
171.1-i |
$6$ |
$8$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{8} \cdot 19^{2} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$18.84609635$ |
1.080897756 |
\( \frac{115714886617}{1539} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -102\) , \( 385\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-102{x}+385$ |
| 171.1-i6 |
171.1-i |
$6$ |
$8$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( - 3^{10} \cdot 19 \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$9.423048177$ |
1.080897756 |
\( \frac{80836867580206264835}{124659} a + \frac{18545249299658286056}{6561} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -175 a - 862\) , \( 2828 a + 12735\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-175a-862\right){x}+2828a+12735$ |
| 171.1-j1 |
171.1-j |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{4} \cdot 19^{2} \) |
$2.81705$ |
$(-a-4), (-a+4), (a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$1.194156810$ |
$3.679988085$ |
8.065308033 |
\( -\frac{1404928}{171} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( -2\) , \( -7\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}-2{x}-7$ |
*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.
