| 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 |
| 9.1-a1 |
9.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$0.53615$ |
$(a)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$3.283496869$ |
0.473931950 |
\( -44330496 a + 76771008 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 25 a - 45\) , \( 72 a - 127\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(25a-45\right){x}+72a-127$ |
| 9.1-a2 |
9.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$0.53615$ |
$(a)$ |
$0$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$29.55147182$ |
0.473931950 |
\( -44330496 a + 76771008 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 25 a - 45\) , \( -117 a + 202\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(25a-45\right){x}-117a+202$ |
| 9.1-a3 |
9.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{3} \) |
$0.53615$ |
$(a)$ |
$0$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \) |
$1$ |
$29.55147182$ |
0.473931950 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}$ |
| 9.1-a4 |
9.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{3} \) |
$0.53615$ |
$(a)$ |
$0$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \) |
$1$ |
$29.55147182$ |
0.473931950 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -7 a + 13\) , \( 6 a - 10\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-7a+13\right){x}+6a-10$ |
| 9.1-a5 |
9.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$0.53615$ |
$(a)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$3.283496869$ |
0.473931950 |
\( 44330496 a + 76771008 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 88 a - 152\) , \( 564 a - 977\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(88a-152\right){x}+564a-977$ |
| 9.1-a6 |
9.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$0.53615$ |
$(a)$ |
$0$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-36$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$29.55147182$ |
0.473931950 |
\( 44330496 a + 76771008 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 88 a - 152\) , \( -717 a + 1242\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(88a-152\right){x}-717a+1242$ |
| 16.1-a1 |
16.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$0.61910$ |
$(a+1)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$17.69503190$ |
0.638514464 |
\( 0 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 1\) , \( -3 a - 5\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+{x}-3a-5$ |
| 16.1-a2 |
16.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$0.61910$ |
$(a+1)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$17.69503190$ |
0.638514464 |
\( 0 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 1\) , \( 3 a + 5\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+{x}+3a+5$ |
| 16.1-a3 |
16.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$0.61910$ |
$(a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$8.847515954$ |
0.638514464 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 4 a - 13\) , \( 11 a - 21\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a-13\right){x}+11a-21$ |
| 16.1-a4 |
16.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$0.61910$ |
$(a+1)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$35.39006381$ |
0.638514464 |
\( -818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 4 a - 13\) , \( -12 a + 19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(4a-13\right){x}-12a+19$ |
| 16.1-a5 |
16.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$0.61910$ |
$(a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$35.39006381$ |
0.638514464 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -a - 3\) , \( -1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-3\right){x}-1$ |
| 16.1-a6 |
16.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$0.61910$ |
$(a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$35.39006381$ |
0.638514464 |
\( 54000 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -a - 3\) , \( -a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-a-3\right){x}-a-1$ |
| 16.1-a7 |
16.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$0.61910$ |
$(a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$8.847515954$ |
0.638514464 |
\( 818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -6 a - 13\) , \( -12 a - 21\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-6a-13\right){x}-12a-21$ |
| 16.1-a8 |
16.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$0.61910$ |
$(a+1)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$35.39006381$ |
0.638514464 |
\( 818626500 a + 1417905000 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -6 a - 13\) , \( 11 a + 19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a-13\right){x}+11a+19$ |
| 22.1-a1 |
22.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2 \cdot 11^{5} \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
$0$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$19.48623858$ |
0.625021394 |
\( -\frac{3800943658260597}{322102} a - \frac{6583595299744607}{322102} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 14261 a - 24701\) , \( -1229141 a + 2128933\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(14261a-24701\right){x}-1229141a+2128933$ |
| 22.1-a2 |
22.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{3} \cdot 11^{15} \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$2.165137620$ |
0.625021394 |
\( -\frac{14621235235888115443}{16708992677662604} a - \frac{20728089694692551503}{16708992677662604} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -23694 a + 41039\) , \( -5992615 a + 10379512\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-23694a+41039\right){x}-5992615a+10379512$ |
| 22.1-a3 |
22.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11^{3} \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$2.165137620$ |
0.625021394 |
\( \frac{7452136447}{340736} a - \frac{12920117437}{340736} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 18 a - 33\) , \( 83 a - 145\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(18a-33\right){x}+83a-145$ |
| 22.1-a4 |
22.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11 \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
$0$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$19.48623858$ |
0.625021394 |
\( -\frac{26727}{88} a + \frac{49507}{88} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -2 a + 2\) , \( -a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a+2\right){x}-a+1$ |
| 22.1-b1 |
22.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2 \cdot 11^{5} \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.1.2 |
$1$ |
\( 5 \) |
$1$ |
$0.597897022$ |
0.862990016 |
\( -\frac{3800943658260597}{322102} a - \frac{6583595299744607}{322102} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 14261 a - 24702\) , \( 1229140 a - 2128935\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(14261a-24702\right){x}+1229140a-2128935$ |
| 22.1-b2 |
22.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{3} \cdot 11^{15} \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
$0$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.1.2 |
$1$ |
\( 3^{2} \cdot 5 \) |
$1$ |
$0.597897022$ |
0.862990016 |
\( -\frac{14621235235888115443}{16708992677662604} a - \frac{20728089694692551503}{16708992677662604} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -23694 a + 41038\) , \( 5992614 a - 10379514\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-23694a+41038\right){x}+5992614a-10379514$ |
| 22.1-b3 |
22.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11^{3} \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
$0$ |
$\Z/15\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.1.1 |
$1$ |
\( 3^{2} \cdot 5 \) |
$1$ |
$14.94742555$ |
0.862990016 |
\( \frac{7452136447}{340736} a - \frac{12920117437}{340736} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 20 a - 32\) , \( -64 a + 112\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(20a-32\right){x}-64a+112$ |
| 22.1-b4 |
22.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11 \) |
$0.67040$ |
$(a+1), (-2a+1)$ |
$0$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$14.94742555$ |
0.862990016 |
\( -\frac{26727}{88} a + \frac{49507}{88} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 3\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+3{x}+1$ |
| 22.2-a1 |
22.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2 \cdot 11^{5} \) |
$0.67040$ |
$(a+1), (2a+1)$ |
$0$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$19.48623858$ |
0.625021394 |
\( \frac{3800943658260597}{322102} a - \frac{6583595299744607}{322102} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 1813 a - 3142\) , \( -55168 a + 95554\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(1813a-3142\right){x}-55168a+95554$ |
| 22.2-a2 |
22.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11^{3} \) |
$0.67040$ |
$(a+1), (2a+1)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$2.165137620$ |
0.625021394 |
\( -\frac{7452136447}{340736} a - \frac{12920117437}{340736} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -18 a - 33\) , \( -83 a - 145\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-18a-33\right){x}-83a-145$ |
| 22.2-a3 |
22.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{3} \cdot 11^{15} \) |
$0.67040$ |
$(a+1), (2a+1)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$2.165137620$ |
0.625021394 |
\( \frac{14621235235888115443}{16708992677662604} a - \frac{20728089694692551503}{16708992677662604} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 1688 a - 2922\) , \( -63194 a + 109458\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(1688a-2922\right){x}-63194a+109458$ |
| 22.2-a4 |
22.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11 \) |
$0.67040$ |
$(a+1), (2a+1)$ |
$0$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$19.48623858$ |
0.625021394 |
\( \frac{26727}{88} a + \frac{49507}{88} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 2 a + 2\) , \( a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+2\right){x}+a+1$ |
| 22.2-b1 |
22.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2 \cdot 11^{5} \) |
$0.67040$ |
$(a+1), (2a+1)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.1.2 |
$1$ |
\( 5 \) |
$1$ |
$0.597897022$ |
0.862990016 |
\( \frac{3800943658260597}{322102} a - \frac{6583595299744607}{322102} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 1813 a - 3143\) , \( 55168 a - 95555\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(1813a-3143\right){x}+55168a-95555$ |
| 22.2-b2 |
22.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{15} \cdot 11^{3} \) |
$0.67040$ |
$(a+1), (2a+1)$ |
$0$ |
$\Z/15\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.1.1 |
$1$ |
\( 3^{2} \cdot 5 \) |
$1$ |
$14.94742555$ |
0.862990016 |
\( -\frac{7452136447}{340736} a - \frac{12920117437}{340736} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -20 a - 32\) , \( 64 a + 112\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-20a-32\right){x}+64a+112$ |
| 22.2-b3 |
22.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{3} \cdot 11^{15} \) |
$0.67040$ |
$(a+1), (2a+1)$ |
$0$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.1, 5B.1.2 |
$1$ |
\( 3^{2} \cdot 5 \) |
$1$ |
$0.597897022$ |
0.862990016 |
\( \frac{14621235235888115443}{16708992677662604} a - \frac{20728089694692551503}{16708992677662604} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 1688 a - 2923\) , \( 63194 a - 109459\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(1688a-2923\right){x}+63194a-109459$ |
| 22.2-b4 |
22.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
22.2 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11 \) |
$0.67040$ |
$(a+1), (2a+1)$ |
$0$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B.1.2, 5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$14.94742555$ |
0.862990016 |
\( \frac{26727}{88} a + \frac{49507}{88} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 3\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+3{x}+1$ |
| 24.1-a1 |
24.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( - 2^{10} \cdot 3 \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$18.60223895$ |
0.671250479 |
\( -\frac{79558124472974}{3} a + 45932904578280 \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 1035 a - 1791\) , \( -23450 a + 40617\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(1035a-1791\right){x}-23450a+40617$ |
| 24.1-a2 |
24.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.325279868$ |
0.671250479 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -15 a + 29\) , \( 322 a - 557\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-15a+29\right){x}+322a-557$ |
| 24.1-a3 |
24.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$18.60223895$ |
0.671250479 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -38 a + 66\) , \( -168 a + 291\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-38a+66\right){x}-168a+291$ |
| 24.1-a4 |
24.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$37.20447790$ |
0.671250479 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 5 a - 6\) , \( -3 a + 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(5a-6\right){x}-3a+6$ |
| 24.1-a5 |
24.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$9.301119475$ |
0.671250479 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 25 a - 41\) , \( 92 a - 159\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(25a-41\right){x}+92a-159$ |
| 24.1-a6 |
24.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$37.20447790$ |
0.671250479 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 65 a - 111\) , \( -348 a + 603\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(65a-111\right){x}-348a+603$ |
| 24.1-a7 |
24.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.325279868$ |
0.671250479 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 385 a - 671\) , \( 5582 a - 9681\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(385a-671\right){x}+5582a-9681$ |
| 24.1-a8 |
24.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( - 2^{10} \cdot 3 \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$18.60223895$ |
0.671250479 |
\( \frac{79558124472974}{3} a + 45932904578280 \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 55 a - 111\) , \( -406 a + 717\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(55a-111\right){x}-406a+717$ |
| 24.1-b1 |
24.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( - 2^{10} \cdot 3 \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$1.420877129$ |
0.820343793 |
\( -\frac{79558124472974}{3} a + 45932904578280 \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 1033 a - 1794\) , \( 24484 a - 42410\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(1033a-1794\right){x}+24484a-42410$ |
| 24.1-b2 |
24.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$5.683508517$ |
0.820343793 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -17 a + 26\) , \( -338 a + 584\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-17a+26\right){x}-338a+584$ |
| 24.1-b3 |
24.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$11.36701703$ |
0.820343793 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -38 a + 66\) , \( 168 a - 291\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-38a+66\right){x}+168a-291$ |
| 24.1-b4 |
24.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$22.73403407$ |
0.820343793 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 3 a - 9\) , \( 7 a - 14\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(3a-9\right){x}+7a-14$ |
| 24.1-b5 |
24.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/2\Z\oplus\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$22.73403407$ |
0.820343793 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 23 a - 44\) , \( -68 a + 116\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(23a-44\right){x}-68a+116$ |
| 24.1-b6 |
24.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$5.683508517$ |
0.820343793 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 63 a - 114\) , \( 412 a - 716\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(63a-114\right){x}+412a-716$ |
| 24.1-b7 |
24.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$22.73403407$ |
0.820343793 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 383 a - 674\) , \( -5198 a + 9008\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(383a-674\right){x}-5198a+9008$ |
| 24.1-b8 |
24.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( - 2^{10} \cdot 3 \) |
$0.68514$ |
$(a+1), (a)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$1.420877129$ |
0.820343793 |
\( \frac{79558124472974}{3} a + 45932904578280 \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 53 a - 114\) , \( 460 a - 830\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(53a-114\right){x}+460a-830$ |
| 33.1-a1 |
33.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3 \cdot 11^{10} \) |
$0.74192$ |
$(a), (2a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$8.787380279$ |
1.268349092 |
\( -\frac{1081911102879025664}{77812273803} a - \frac{605477717460973120}{25937424601} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 22986 a - 39809\) , \( -2497992 a + 4326651\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(22986a-39809\right){x}-2497992a+4326651$ |
| 33.1-a2 |
33.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{5} \cdot 11^{2} \) |
$0.74192$ |
$(a), (2a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$8.787380279$ |
1.268349092 |
\( -\frac{2084278784}{3267} a + \frac{1204895680}{1089} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 6 a - 8\) , \( 12 a - 19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(6a-8\right){x}+12a-19$ |
| 33.1-a3 |
33.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{10} \cdot 11 \) |
$0.74192$ |
$(a), (2a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$8.787380279$ |
1.268349092 |
\( \frac{2291200}{2673} a + \frac{1654208}{2673} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( 15 a - 28\) , \( 96 a - 167\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(15a-28\right){x}+96a-167$ |
| 33.1-a4 |
33.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{2} \cdot 11^{5} \) |
$0.74192$ |
$(a), (2a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$8.787380279$ |
1.268349092 |
\( \frac{313724549420617141760}{483153} a + \frac{543386859178009155008}{483153} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 57891 a - 100269\) , \( -25338895 a + 43888254\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(57891a-100269\right){x}-25338895a+43888254$ |
*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.
