| 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 |
| 9.2-a1 |
9.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{3} \) |
$1.34929$ |
$(-a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$29.55147182$ |
1.694893148 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 2 a + 32\) , \( 9 a + 29\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+32\right){x}+9a+29$ |
| 9.2-a2 |
9.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{3} \) |
$1.34929$ |
$(-a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$29.55147182$ |
1.694893148 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 87 a + 381\) , \( 349 a + 1521\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(87a+381\right){x}+349a+1521$ |
| 9.2-b1 |
9.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{10} \) |
$1.34929$ |
$(-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1.105793423$ |
$6.013673307$ |
3.051174382 |
\( -\frac{53248}{81} a - \frac{131072}{81} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -4759 a + 20744\) , \( 231623 a - 1009626\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-4759a+20744\right){x}+231623a-1009626$ |
| 9.2-c1 |
9.2-c |
$2$ |
$19$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.34929$ |
$(-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1.624922199$ |
$2.325972790$ |
1.734164921 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -16 a - 70\) , \( -73 a - 323\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-16a-70\right){x}-73a-323$ |
| 9.2-c2 |
9.2-c |
$2$ |
$19$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.34929$ |
$(-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.085522221$ |
$44.19348301$ |
1.734164921 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -16 a - 70\) , \( 73 a + 318\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-16a-70\right){x}+73a+318$ |
| 9.2-d1 |
9.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{10} \) |
$1.34929$ |
$(-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.098423053$ |
$19.00729698$ |
0.858361815 |
\( -\frac{53248}{81} a - \frac{131072}{81} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -4759 a + 20744\) , \( -231623 a + 1009621\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-4759a+20744\right){x}-231623a+1009621$ |
| 9.3-a1 |
9.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{3} \) |
$1.34929$ |
$(-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$29.55147182$ |
1.694893148 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -76 a + 372\) , \( 23 a - 32\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-76a+372\right){x}+23a-32$ |
| 9.3-a2 |
9.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{3} \) |
$1.34929$ |
$(-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$29.55147182$ |
1.694893148 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 9 a + 41\) , \( 23 a + 100\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(9a+41\right){x}+23a+100$ |
| 9.3-b1 |
9.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{10} \) |
$1.34929$ |
$(-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1.105793423$ |
$6.013673307$ |
3.051174382 |
\( \frac{53248}{81} a - \frac{131072}{81} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 4759 a + 20744\) , \( -231623 a - 1009626\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(4759a+20744\right){x}-231623a-1009626$ |
| 9.3-c1 |
9.3-c |
$2$ |
$19$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.34929$ |
$(-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1.624922199$ |
$2.325972790$ |
1.734164921 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 16 a - 70\) , \( 73 a - 323\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(16a-70\right){x}+73a-323$ |
| 9.3-c2 |
9.3-c |
$2$ |
$19$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.34929$ |
$(-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.085522221$ |
$44.19348301$ |
1.734164921 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 16 a - 70\) , \( -73 a + 318\bigr] \) |
${y}^2+{y}={x}^{3}+\left(16a-70\right){x}-73a+318$ |
| 9.3-d1 |
9.3-d |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{10} \) |
$1.34929$ |
$(-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.098423053$ |
$19.00729698$ |
0.858361815 |
\( \frac{53248}{81} a - \frac{131072}{81} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 4759 a + 20744\) , \( 231623 a + 1009621\bigr] \) |
${y}^2+{y}={x}^{3}+\left(4759a+20744\right){x}+231623a+1009621$ |
| 16.1-a1 |
16.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{18} \) |
$1.55803$ |
$(-3a+13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$9.924776877$ |
2.276899970 |
\( -\frac{27}{8} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 6 a + 26\) , \( 8 a + 36\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6a+26\right){x}+8a+36$ |
| 16.1-b1 |
16.1-b |
$2$ |
$19$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$1.55803$ |
$(-3a+13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$8.780354734$ |
1.007175762 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -8\) , \( -2 a\bigr] \) |
${y}^2={x}^{3}-8{x}-2a$ |
| 16.1-b2 |
16.1-b |
$2$ |
$19$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$1.55803$ |
$(-3a+13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$8.780354734$ |
1.007175762 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -8\) , \( 2 a\bigr] \) |
${y}^2={x}^{3}-8{x}+2a$ |
| 16.1-c1 |
16.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{18} \) |
$1.55803$ |
$(-3a+13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$9.924776877$ |
2.276899970 |
\( -\frac{27}{8} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 5 a + 16\) , \( 8 a + 31\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(5a+16\right){x}+8a+31$ |
| 18.1-a1 |
18.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{14} \cdot 3^{12} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$5.365735541$ |
2.154222274 |
\( -\frac{81634634531}{22674816} a + \frac{204402908939}{11337408} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -6112 a - 26627\) , \( -275601 a - 1201300\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6112a-26627\right){x}-275601a-1201300$ |
| 18.1-a2 |
18.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( - 2^{7} \cdot 3^{24} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$2.682867770$ |
2.154222274 |
\( \frac{3015980695953593}{502096953744} a + \frac{13147717480342991}{502096953744} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -46792 a - 203947\) , \( 11319975 a + 49342644\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-46792a-203947\right){x}+11319975a+49342644$ |
| 18.1-b1 |
18.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( - 2^{7} \cdot 3^{24} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$2.682867770$ |
2.154222274 |
\( -\frac{3015980695953593}{502096953744} a + \frac{13147717480342991}{502096953744} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 46791 a - 203947\) , \( -11319976 a + 49342644\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(46791a-203947\right){x}-11319976a+49342644$ |
| 18.1-b2 |
18.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{14} \cdot 3^{12} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$5.365735541$ |
2.154222274 |
\( \frac{81634634531}{22674816} a + \frac{204402908939}{11337408} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 6111 a - 26627\) , \( 275600 a - 1201300\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(6111a-26627\right){x}+275600a-1201300$ |
| 18.1-c1 |
18.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{14} \cdot 3^{12} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 11 \) |
$0.134555585$ |
$10.28817665$ |
1.746731012 |
\( -\frac{81634634531}{22674816} a + \frac{204402908939}{11337408} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -6110 a - 26622\) , \( 269490 a + 1174670\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-6110a-26622\right){x}+269490a+1174670$ |
| 18.1-c2 |
18.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( - 2^{7} \cdot 3^{24} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 11 \) |
$0.269111170$ |
$2.572044164$ |
1.746731012 |
\( \frac{3015980695953593}{502096953744} a + \frac{13147717480342991}{502096953744} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -46790 a - 203942\) , \( -11366766 a - 49546594\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-46790a-203942\right){x}-11366766a-49546594$ |
| 18.1-d1 |
18.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( - 2^{7} \cdot 3^{24} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 11 \) |
$0.269111170$ |
$2.572044164$ |
1.746731012 |
\( -\frac{3015980695953593}{502096953744} a + \frac{13147717480342991}{502096953744} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 46789 a - 203942\) , \( 11366766 a - 49546594\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(46789a-203942\right){x}+11366766a-49546594$ |
| 18.1-d2 |
18.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{14} \cdot 3^{12} \) |
$1.60459$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 11 \) |
$0.134555585$ |
$10.28817665$ |
1.746731012 |
\( \frac{81634634531}{22674816} a + \frac{204402908939}{11337408} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 6109 a - 26622\) , \( -269490 a + 1174670\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(6109a-26622\right){x}-269490a+1174670$ |
| 18.2-a1 |
18.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.2 |
\( 2 \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \) |
$1.60459$ |
$(-3a+13), (-a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$6.303728707$ |
4.338523642 |
\( -\frac{27}{8} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -a - 2\) , \( -9 a - 44\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-a-2\right){x}-9a-44$ |
| 18.2-b1 |
18.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.2 |
\( 2 \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \) |
$1.60459$ |
$(-3a+13), (-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$0.123958993$ |
$20.83448291$ |
1.184988031 |
\( -\frac{27}{8} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -a - 7\) , \( 8 a + 34\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-a-7\right){x}+8a+34$ |
| 18.3-a1 |
18.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.3 |
\( 2 \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \) |
$1.60459$ |
$(-3a+13), (-a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$6.303728707$ |
4.338523642 |
\( -\frac{27}{8} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -2\) , \( 9 a - 44\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}-2{x}+9a-44$ |
| 18.3-b1 |
18.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
18.3 |
\( 2 \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \) |
$1.60459$ |
$(-3a+13), (-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$0.123958993$ |
$20.83448291$ |
1.184988031 |
\( -\frac{27}{8} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -7\) , \( -9 a + 34\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}-7{x}-9a+34$ |
| 19.1-a1 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$1.62642$ |
$(a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$17.03289160$ |
3.907613328 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -769\) , \( 8465\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}-769{x}+8465$ |
| 19.1-a2 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$1.62642$ |
$(a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$1$ |
$17.03289160$ |
3.907613328 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -9\) , \( 10\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}-9{x}+10$ |
| 19.1-a3 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$1.62642$ |
$(a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$17.03289160$ |
3.907613328 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( 1\) , \( -5\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+{x}-5$ |
| 19.1-b1 |
19.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$1.62642$ |
$(a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$16.54388095$ |
$0.205438503$ |
1.559453518 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -769\) , \( -8470\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-769{x}-8470$ |
| 19.1-b2 |
19.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$1.62642$ |
$(a)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$5.514626985$ |
$1.848946532$ |
1.559453518 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -9\) , \( -15\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-9{x}-15$ |
| 19.1-b3 |
19.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$1.62642$ |
$(a)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1.838208995$ |
$16.64051879$ |
1.559453518 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+{x}$ |
| 20.1-a1 |
20.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$1.64741$ |
$(-3a+13), (2a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$4.815250970$ |
$1.424343704$ |
3.146928843 |
\( -\frac{842348523228}{15625} a - \frac{3671711480264}{15625} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -10255 a - 44704\) , \( -1216748 a - 5303685\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-10255a-44704\right){x}-1216748a-5303685$ |
| 20.1-a2 |
20.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$1.64741$ |
$(-3a+13), (2a+9)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.605083656$ |
$12.81909334$ |
3.146928843 |
\( -\frac{612}{25} a - \frac{3656}{25} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -85 a - 374\) , \( -2930 a - 12775\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-85a-374\right){x}-2930a-12775$ |
| 20.1-b1 |
20.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$1.64741$ |
$(-3a+13), (2a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$0.084888632$ |
$14.05008960$ |
1.641735093 |
\( -\frac{842348523228}{15625} a - \frac{3671711480264}{15625} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -10254 a - 44701\) , \( 1141271 a + 4974682\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-10254a-44701\right){x}+1141271a+4974682$ |
| 20.1-b2 |
20.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$1.64741$ |
$(-3a+13), (2a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.254665897$ |
$14.05008960$ |
1.641735093 |
\( -\frac{612}{25} a - \frac{3656}{25} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -84 a - 371\) , \( 2293 a + 9992\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-84a-371\right){x}+2293a+9992$ |
| 20.2-a1 |
20.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$1.64741$ |
$(-3a+13), (-2a+9)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.605083656$ |
$12.81909334$ |
3.146928843 |
\( \frac{612}{25} a - \frac{3656}{25} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 92 a - 364\) , \( 2556 a - 11098\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(92a-364\right){x}+2556a-11098$ |
| 20.2-a2 |
20.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$1.64741$ |
$(-3a+13), (-2a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$4.815250970$ |
$1.424343704$ |
3.146928843 |
\( \frac{842348523228}{15625} a - \frac{3671711480264}{15625} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 10262 a - 44694\) , \( 1172044 a - 5108778\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(10262a-44694\right){x}+1172044a-5108778$ |
| 20.2-b1 |
20.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$1.64741$ |
$(-3a+13), (-2a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.254665897$ |
$14.05008960$ |
1.641735093 |
\( \frac{612}{25} a - \frac{3656}{25} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 93 a - 361\) , \( -2664 a + 11669\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(93a-361\right){x}-2664a+11669$ |
| 20.2-b2 |
20.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$1.64741$ |
$(-3a+13), (-2a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$0.084888632$ |
$14.05008960$ |
1.641735093 |
\( \frac{842348523228}{15625} a - \frac{3671711480264}{15625} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 10263 a - 44691\) , \( -1185972 a + 5169589\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(10263a-44691\right){x}-1185972a+5169589$ |
| 24.1-a1 |
24.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{14} \) |
$1.72424$ |
$(-3a+13), (-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \cdot 7 \) |
$0.060578918$ |
$5.956989719$ |
2.318086285 |
\( \frac{2379806720}{4782969} a - \frac{24019195904}{4782969} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( -52701 a - 229718\) , \( -17439999 a - 76019196\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-52701a-229718\right){x}-17439999a-76019196$ |
| 24.1-b1 |
24.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$1.72424$ |
$(-3a+13), (-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.049960597$ |
$24.93547084$ |
1.143216247 |
\( \frac{2048}{9} a - \frac{2048}{9} \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( a + 2\) , \( -3 a + 9\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(a+2\right){x}-3a+9$ |
| 24.1-c1 |
24.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{14} \) |
$1.72424$ |
$(-3a+13), (-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.627687552$ |
$5.423983166$ |
3.124244690 |
\( \frac{2379806720}{4782969} a - \frac{24019195904}{4782969} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( -52701 a - 229718\) , \( 17439998 a + 76019186\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-52701a-229718\right){x}+17439998a+76019186$ |
| 24.1-d1 |
24.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$1.72424$ |
$(-3a+13), (-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.286981021$ |
$13.11618382$ |
3.454171233 |
\( \frac{2048}{9} a - \frac{2048}{9} \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( a + 2\) , \( 2 a - 19\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(a+2\right){x}+2a-19$ |
| 24.2-a1 |
24.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
24.2 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{14} \) |
$1.72424$ |
$(-3a+13), (-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \cdot 7 \) |
$0.060578918$ |
$5.956989719$ |
2.318086285 |
\( -\frac{2379806720}{4782969} a - \frac{24019195904}{4782969} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( 52701 a - 229718\) , \( 17439998 a - 76019196\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(52701a-229718\right){x}+17439998a-76019196$ |
| 24.2-b1 |
24.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
24.2 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$1.72424$ |
$(-3a+13), (-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.049960597$ |
$24.93547084$ |
1.143216247 |
\( -\frac{2048}{9} a - \frac{2048}{9} \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( -a + 2\) , \( 2 a + 9\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-a+2\right){x}+2a+9$ |
| 24.2-c1 |
24.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
24.2 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{14} \) |
$1.72424$ |
$(-3a+13), (-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.627687552$ |
$5.423983166$ |
3.124244690 |
\( -\frac{2379806720}{4782969} a - \frac{24019195904}{4782969} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( 52701 a - 229718\) , \( -17439999 a + 76019186\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(52701a-229718\right){x}-17439999a+76019186$ |
| 24.2-d1 |
24.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
24.2 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$1.72424$ |
$(-3a+13), (-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.286981021$ |
$13.11618382$ |
3.454171233 |
\( -\frac{2048}{9} a - \frac{2048}{9} \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( -a + 2\) , \( -3 a - 19\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-a+2\right){x}-3a-19$ |
*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.
