| 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 |
| 15.1-a1 |
15.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$16.31232646$ |
0.889910366 |
\( -\frac{721}{75} a - \frac{1}{15} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -a\) , \( -4 a - 7\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}-a{x}-4a-7$ |
| 15.1-a2 |
15.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{4} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.156163233$ |
0.889910366 |
\( -\frac{100981119568896026467}{1875} a + \frac{56373474375475478518}{375} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 174 a + 260\) , \( 145 a + 374\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(174a+260\right){x}+145a+374$ |
| 15.1-a3 |
15.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{8} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$16.31232646$ |
0.889910366 |
\( -\frac{127041323975657}{1171875} a + \frac{70922198887903}{234375} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -41 a - 75\) , \( 73 a + 131\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-41a-75\right){x}+73a+131$ |
| 15.1-a4 |
15.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$16.31232646$ |
0.889910366 |
\( \frac{169820651}{5625} a + \frac{28920482}{375} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 27 a - 62\) , \( -106 a + 305\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(27a-62\right){x}-106a+305$ |
| 15.1-a5 |
15.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{16} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.156163233$ |
0.889910366 |
\( \frac{54809252307092563}{457763671875} a + \frac{17662537283047898}{91552734375} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 382 a - 1057\) , \( -6554 a + 18298\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(382a-1057\right){x}-6554a+18298$ |
| 15.1-a6 |
15.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.078081616$ |
0.889910366 |
\( \frac{11403943879867}{2025} a + \frac{4085550627187}{405} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 67 a - 177\) , \( 271 a - 757\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(67a-177\right){x}+271a-757$ |
| 15.1-b1 |
15.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$19.53039258$ |
1.065470266 |
\( -\frac{721}{75} a - \frac{1}{15} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+{x}$ |
| 15.1-b2 |
15.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{4} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.220649536$ |
1.065470266 |
\( -\frac{100981119568896026467}{1875} a + \frac{56373474375475478518}{375} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 275 a - 734\) , \( 3747 a - 10492\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(275a-734\right){x}+3747a-10492$ |
| 15.1-b3 |
15.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{8} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$4.882598147$ |
1.065470266 |
\( -\frac{127041323975657}{1171875} a + \frac{70922198887903}{234375} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 15 a - 49\) , \( 66 a - 181\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(15a-49\right){x}+66a-181$ |
| 15.1-b4 |
15.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
$0$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$19.53039258$ |
1.065470266 |
\( \frac{169820651}{5625} a + \frac{28920482}{375} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -4\) , \( 3 a - 1\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}-4{x}+3a-1$ |
| 15.1-b5 |
15.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{16} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.220649536$ |
1.065470266 |
\( \frac{54809252307092563}{457763671875} a + \frac{17662537283047898}{91552734375} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -5 a - 84\) , \( -3 a - 310\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a-84\right){x}-3a-310$ |
| 15.1-b6 |
15.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$0.80588$ |
$(-a+2), (-a+1)$ |
$0$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$19.53039258$ |
1.065470266 |
\( \frac{11403943879867}{2025} a + \frac{4085550627187}{405} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -15 a - 39\) , \( 72 a + 135\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-15a-39\right){x}+72a+135$ |
| 15.2-a1 |
15.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{16} \) |
$0.80588$ |
$(-a+2), (-a)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.156163233$ |
0.889910366 |
\( -\frac{54809252307092563}{457763671875} a + \frac{143121938722332053}{457763671875} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -379 a - 681\) , \( 5873 a + 10526\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-379a-681\right){x}+5873a+10526$ |
| 15.2-a2 |
15.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$0.80588$ |
$(-a+2), (-a)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$16.31232646$ |
0.889910366 |
\( \frac{721}{75} a - \frac{242}{25} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( 0\) , \( 3 a - 10\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+3a-10$ |
| 15.2-a3 |
15.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$0.80588$ |
$(-a+2), (-a)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.078081616$ |
0.889910366 |
\( -\frac{11403943879867}{2025} a + \frac{1178951741326}{75} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -64 a - 116\) , \( -387 a - 694\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-64a-116\right){x}-387a-694$ |
| 15.2-a4 |
15.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$0.80588$ |
$(-a+2), (-a)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$16.31232646$ |
0.889910366 |
\( -\frac{169820651}{5625} a + \frac{603627881}{5625} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( -24 a - 41\) , \( 65 a + 116\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-24a-41\right){x}+65a+116$ |
| 15.2-a5 |
15.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{8} \) |
$0.80588$ |
$(-a+2), (-a)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$16.31232646$ |
0.889910366 |
\( \frac{127041323975657}{1171875} a + \frac{75856556821286}{390625} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( 40 a - 115\) , \( -74 a + 205\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(40a-115\right){x}-74a+205$ |
| 15.2-a6 |
15.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{4} \) |
$0.80588$ |
$(-a+2), (-a)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.156163233$ |
0.889910366 |
\( \frac{100981119568896026467}{1875} a + \frac{180886252308481366123}{1875} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -175 a + 435\) , \( -146 a + 520\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-175a+435\right){x}-146a+520$ |
| 15.2-b1 |
15.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{16} \) |
$0.80588$ |
$(-a+2), (-a)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.220649536$ |
1.065470266 |
\( -\frac{54809252307092563}{457763671875} a + \frac{143121938722332053}{457763671875} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 5 a - 90\) , \( 8 a - 403\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(5a-90\right){x}+8a-403$ |
| 15.2-b2 |
15.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$0.80588$ |
$(-a+2), (-a)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$19.53039258$ |
1.065470266 |
\( \frac{721}{75} a - \frac{242}{25} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}$ |
| 15.2-b3 |
15.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$0.80588$ |
$(-a+2), (-a)$ |
$0$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$19.53039258$ |
1.065470266 |
\( -\frac{11403943879867}{2025} a + \frac{1178951741326}{75} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 15 a - 55\) , \( -57 a + 152\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(15a-55\right){x}-57a+152$ |
| 15.2-b4 |
15.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$0.80588$ |
$(-a+2), (-a)$ |
$0$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$19.53039258$ |
1.065470266 |
\( -\frac{169820651}{5625} a + \frac{603627881}{5625} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -5\) , \( -3 a - 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}-5{x}-3a-3$ |
| 15.2-b5 |
15.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{8} \) |
$0.80588$ |
$(-a+2), (-a)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$4.882598147$ |
1.065470266 |
\( \frac{127041323975657}{1171875} a + \frac{75856556821286}{390625} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -15 a - 35\) , \( -81 a - 150\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-15a-35\right){x}-81a-150$ |
| 15.2-b6 |
15.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{4} \) |
$0.80588$ |
$(-a+2), (-a)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.220649536$ |
1.065470266 |
\( \frac{100981119568896026467}{1875} a + \frac{180886252308481366123}{1875} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -275 a - 460\) , \( -4022 a - 7205\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-275a-460\right){x}-4022a-7205$ |
| 16.1-a1 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$0.81899$ |
$(2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2, 7$ |
2B, 7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$17.69503190$ |
0.965343132 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 2\) , \( -2\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+2\right){x}-2$ |
| 16.1-a2 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$0.81899$ |
$(2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2, 7$ |
2B, 7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$17.69503190$ |
0.965343132 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 2\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}$ |
| 16.1-a3 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{16} \) |
$0.81899$ |
$(2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$17.69503190$ |
0.965343132 |
\( 54000 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 6 a - 13\) , \( 11 a - 31\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(6a-13\right){x}+11a-31$ |
| 16.1-a4 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{16} \) |
$0.81899$ |
$(2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$17.69503190$ |
0.965343132 |
\( 54000 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -4 a - 8\) , \( -16 a - 28\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a-8\right){x}-16a-28$ |
| 17.1-a1 |
17.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( -17 \) |
$0.83150$ |
$(-2a+3)$ |
$0$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$45.16448385$ |
1.095077597 |
\( \frac{20811}{17} a - \frac{19591}{17} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -7\) , \( -2 a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}-7{x}-2a+2$ |
| 17.1-a2 |
17.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( - 17^{3} \) |
$0.83150$ |
$(-2a+3)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$5.018275983$ |
1.095077597 |
\( \frac{2723256379739}{4913} a + \frac{4878142779916}{4913} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -15 a + 33\) , \( 10 a - 33\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-15a+33\right){x}+10a-33$ |
| 17.1-b1 |
17.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( -17 \) |
$0.83150$ |
$(-2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.092314733$ |
$19.62917645$ |
0.790848774 |
\( \frac{20811}{17} a - \frac{19591}{17} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -a + 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-a+1\right){x}$ |
| 17.1-b2 |
17.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( - 17^{3} \) |
$0.83150$ |
$(-2a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.276944199$ |
$19.62917645$ |
0.790848774 |
\( \frac{2723256379739}{4913} a + \frac{4878142779916}{4913} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -11 a - 14\) , \( 26 a + 50\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-11a-14\right){x}+26a+50$ |
| 17.2-a1 |
17.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( -17 \) |
$0.83150$ |
$(2a+1)$ |
$0$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$45.16448385$ |
1.095077597 |
\( -\frac{20811}{17} a + \frac{1220}{17} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -2 a - 5\) , \( a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-2a-5\right){x}+a+1$ |
| 17.2-a2 |
17.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( - 17^{3} \) |
$0.83150$ |
$(2a+1)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$5.018275983$ |
1.095077597 |
\( -\frac{2723256379739}{4913} a + \frac{7601399159655}{4913} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 13 a + 20\) , \( -11 a - 22\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(13a+20\right){x}-11a-22$ |
| 17.2-b1 |
17.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( -17 \) |
$0.83150$ |
$(2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.092314733$ |
$19.62917645$ |
0.790848774 |
\( -\frac{20811}{17} a + \frac{1220}{17} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}$ |
| 17.2-b2 |
17.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( - 17^{3} \) |
$0.83150$ |
$(2a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.276944199$ |
$19.62917645$ |
0.790848774 |
\( -\frac{2723256379739}{4913} a + \frac{7601399159655}{4913} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 10 a - 25\) , \( -26 a + 76\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(10a-25\right){x}-26a+76$ |
| 21.1-a1 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{16} \) |
$0.87660$ |
$(-a+2), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.651881942$ |
0.796905972 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 170 a - 477\) , \( -5038 a + 14062\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(170a-477\right){x}-5038a+14062$ |
| 21.1-a2 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{2} \) |
$0.87660$ |
$(-a+2), (a+3)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$14.60752776$ |
0.796905972 |
\( \frac{103823}{63} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -5 a + 13\) , \( -5 a + 13\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-5a+13\right){x}-5a+13$ |
| 21.1-a3 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$0.87660$ |
$(-a+2), (a+3)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$14.60752776$ |
0.796905972 |
\( \frac{7189057}{3969} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 20 a - 57\) , \( -4 a + 10\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(20a-57\right){x}-4a+10$ |
| 21.1-a4 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{16} \cdot 7^{2} \) |
$0.87660$ |
$(-a+2), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.651881942$ |
0.796905972 |
\( \frac{6570725617}{45927} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 195 a - 547\) , \( 2355 a - 6577\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(195a-547\right){x}+2355a-6577$ |
| 21.1-a5 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$0.87660$ |
$(-a+2), (a+3)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$14.60752776$ |
0.796905972 |
\( \frac{13027640977}{21609} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 245 a - 687\) , \( -3019 a + 8425\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(245a-687\right){x}-3019a+8425$ |
| 21.1-a6 |
21.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{4} \) |
$0.87660$ |
$(-a+2), (a+3)$ |
$0$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$14.60752776$ |
0.796905972 |
\( \frac{53297461115137}{147} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 3920 a - 10977\) , \( -200440 a + 559528\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(3920a-10977\right){x}-200440a+559528$ |
| 21.1-b1 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{16} \) |
$0.87660$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.638508823$ |
$0.814020435$ |
0.937376813 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -34\) , \( -217\bigr] \) |
${y}^2+{x}{y}={x}^{3}-34{x}-217$ |
| 21.1-b2 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{2} \) |
$0.87660$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.319254411$ |
$13.02432697$ |
0.937376813 |
\( \frac{103823}{63} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}$ |
| 21.1-b3 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$0.87660$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.659627205$ |
$13.02432697$ |
0.937376813 |
\( \frac{7189057}{3969} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -4\) , \( -1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-4{x}-1$ |
| 21.1-b4 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{16} \cdot 7^{2} \) |
$0.87660$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.329813602$ |
$13.02432697$ |
0.937376813 |
\( \frac{6570725617}{45927} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -39\) , \( 90\bigr] \) |
${y}^2+{x}{y}={x}^{3}-39{x}+90$ |
| 21.1-b5 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$0.87660$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.319254411$ |
$3.256081743$ |
0.937376813 |
\( \frac{13027640977}{21609} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -49\) , \( -136\bigr] \) |
${y}^2+{x}{y}={x}^{3}-49{x}-136$ |
| 21.1-b6 |
21.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{4} \) |
$0.87660$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.638508823$ |
$0.814020435$ |
0.937376813 |
\( \frac{53297461115137}{147} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -784\) , \( -8515\bigr] \) |
${y}^2+{x}{y}={x}^{3}-784{x}-8515$ |
| 25.1-a1 |
25.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{16} \) |
$0.91566$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.961531894$ |
1.082695022 |
\( -\frac{359104782699}{244140625} a - \frac{52148361654}{48828125} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -26 a - 43\) , \( 84 a + 148\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-26a-43\right){x}+84a+148$ |
| 25.1-a2 |
25.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{16} \) |
$0.91566$ |
$(-a), (-a+1)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.961531894$ |
1.082695022 |
\( \frac{359104782699}{244140625} a - \frac{619846590969}{244140625} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 29 a - 66\) , \( -125 a + 369\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(29a-66\right){x}-125a+369$ |
*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.
