| 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 |
| 768.1-a1 |
768.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{14} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$4.829615067$ |
3.688679437 |
\( -\frac{11855696}{2187} a + \frac{117012512}{2187} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -31 a - 77\) , \( 157 a + 258\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-31a-77\right){x}+157a+258$ |
| 768.1-a2 |
768.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{8} \cdot 3^{7} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 7 \) |
$1$ |
$9.659230135$ |
3.688679437 |
\( \frac{25019564800}{81} a + \frac{44817242368}{81} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -36 a - 62\) , \( 155 a + 275\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-36a-62\right){x}+155a+275$ |
| 768.1-b1 |
768.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$4.365994989$ |
0.952738215 |
\( -\frac{29376256}{3} a + \frac{81893120}{3} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -8 a - 13\) , \( -14 a - 26\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8a-13\right){x}-14a-26$ |
| 768.1-b2 |
768.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.365994989$ |
0.952738215 |
\( \frac{827842288}{3} a + \frac{1482926240}{3} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -128 a - 228\) , \( -1116 a - 2000\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-128a-228\right){x}-1116a-2000$ |
| 768.1-c1 |
768.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{8} \cdot 3^{7} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$2.642349772$ |
$5.062045892$ |
1.459408060 |
\( -\frac{25019564800}{81} a + \frac{69836807168}{81} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( a - 5\) , \( 48 a + 81\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(a-5\right){x}+48a+81$ |
| 768.1-c2 |
768.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{14} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.321174886$ |
$5.062045892$ |
1.459408060 |
\( \frac{11855696}{2187} a + \frac{35052272}{729} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -119 a - 220\) , \( 1117 a + 1996\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-119a-220\right){x}+1117a+1996$ |
| 768.1-d1 |
768.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{16} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$5.060144339$ |
2.208428044 |
\( -\frac{14303296}{3} a + \frac{39925904}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 4 a + 4\) , \( -8 a - 20\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(4a+4\right){x}-8a-20$ |
| 768.1-d2 |
768.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$20.24057735$ |
2.208428044 |
\( \frac{16384}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -a - 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-a-1\right){x}$ |
| 768.1-d3 |
768.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$20.24057735$ |
2.208428044 |
\( \frac{109744}{9} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -6 a - 11\) , \( 19 a + 34\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-6a-11\right){x}+19a+34$ |
| 768.1-d4 |
768.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{16} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$5.060144339$ |
2.208428044 |
\( \frac{14303296}{3} a + \frac{25622608}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -21 a - 36\) , \( -61 a - 109\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-21a-36\right){x}-61a-109$ |
| 768.1-e1 |
768.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{16} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.941115895$ |
$18.30323376$ |
2.936772644 |
\( -\frac{14303296}{3} a + \frac{39925904}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 4 a + 4\) , \( 8 a + 20\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(4a+4\right){x}+8a+20$ |
| 768.1-e2 |
768.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1.470557947$ |
$36.60646753$ |
2.936772644 |
\( \frac{16384}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -a - 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-a-1\right){x}$ |
| 768.1-e3 |
768.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.735278973$ |
$9.151616883$ |
2.936772644 |
\( \frac{109744}{9} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -6 a - 11\) , \( -19 a - 34\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-6a-11\right){x}-19a-34$ |
| 768.1-e4 |
768.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{16} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.735278973$ |
$18.30323376$ |
2.936772644 |
\( \frac{14303296}{3} a + \frac{25622608}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -21 a - 36\) , \( 61 a + 109\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-21a-36\right){x}+61a+109$ |
| 768.1-f1 |
768.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$40.83601149$ |
2.227787068 |
\( -\frac{29376256}{3} a + \frac{81893120}{3} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -8 a - 13\) , \( 14 a + 26\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-8a-13\right){x}+14a+26$ |
| 768.1-f2 |
768.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$20.41800574$ |
2.227787068 |
\( \frac{827842288}{3} a + \frac{1482926240}{3} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -128 a - 228\) , \( 1116 a + 2000\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-128a-228\right){x}+1116a+2000$ |
| 768.1-g1 |
768.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{20} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$3.410424115$ |
2.976862221 |
\( -\frac{5202690124}{3} a + \frac{14522206328}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 47 a + 81\) , \( 725 a + 1295\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(47a+81\right){x}+725a+1295$ |
| 768.1-g2 |
768.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$13.64169646$ |
2.976862221 |
\( \frac{4864}{3} a + \frac{8704}{3} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 1\) , \( 4 a - 11\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+{x}+4a-11$ |
| 768.1-g3 |
768.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$13.64169646$ |
2.976862221 |
\( -\frac{53200}{3} a + 58128 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 25 a - 69\) , \( 81 a - 226\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(25a-69\right){x}+81a-226$ |
| 768.1-g4 |
768.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{20} \cdot 3^{4} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$13.64169646$ |
2.976862221 |
\( \frac{121322980}{9} a + \frac{217342184}{9} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 45 a - 129\) , \( -135 a + 378\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(45a-129\right){x}-135a+378$ |
| 768.1-h1 |
768.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.360229675$ |
2.775831802 |
\( \frac{1456}{3} a - 112 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( a - 1\) , \( a - 2\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(a-1\right){x}+a-2$ |
| 768.1-h2 |
768.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{20} \cdot 3^{4} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$6.360229675$ |
2.775831802 |
\( -\frac{5095532}{9} a + \frac{14244392}{9} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 21 a - 61\) , \( 65 a - 182\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(21a-61\right){x}+65a-182$ |
| 768.1-i1 |
768.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1.294261724$ |
$22.72318029$ |
3.208865982 |
\( \frac{256}{3} a + \frac{4864}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -a + 3\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-a+3\right){x}$ |
| 768.1-i2 |
768.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.647130862$ |
$22.72318029$ |
3.208865982 |
\( \frac{77872}{3} a + \frac{145568}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 4 a - 12\) , \( -4 a + 12\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(4a-12\right){x}-4a+12$ |
| 768.1-j1 |
768.1-j |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.433845102$ |
$10.88906985$ |
4.123593315 |
\( \frac{2000}{9} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 2 a + 5\) , \( 3 a + 6\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(2a+5\right){x}+3a+6$ |
| 768.1-j2 |
768.1-j |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{20} \cdot 3^{8} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.216922551$ |
$10.88906985$ |
4.123593315 |
\( \frac{665500}{81} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -18 a - 35\) , \( 79 a + 142\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-18a-35\right){x}+79a+142$ |
| 768.1-k1 |
768.1-k |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$20.41800574$ |
2.227787068 |
\( -\frac{827842288}{3} a + 770256176 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 128 a - 356\) , \( -1116 a + 3116\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(128a-356\right){x}-1116a+3116$ |
| 768.1-k2 |
768.1-k |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$40.83601149$ |
2.227787068 |
\( \frac{29376256}{3} a + \frac{52516864}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 8 a - 21\) , \( -14 a + 40\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(8a-21\right){x}-14a+40$ |
| 768.1-l1 |
768.1-l |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.86979328$ |
2.808419138 |
\( \frac{1456}{3} a - 112 \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( a - 1\) , \( -a + 2\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(a-1\right){x}-a+2$ |
| 768.1-l2 |
768.1-l |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{20} \cdot 3^{4} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.86979328$ |
2.808419138 |
\( -\frac{5095532}{9} a + \frac{14244392}{9} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 21 a - 61\) , \( -65 a + 182\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(21a-61\right){x}-65a+182$ |
| 768.1-m1 |
768.1-m |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.408489382$ |
1.180229142 |
\( \frac{2000}{9} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 2 a + 5\) , \( -3 a - 6\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(2a+5\right){x}-3a-6$ |
| 768.1-m2 |
768.1-m |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{20} \cdot 3^{8} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.408489382$ |
1.180229142 |
\( \frac{665500}{81} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -18 a - 35\) , \( -79 a - 142\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-18a-35\right){x}-79a-142$ |
| 768.1-n1 |
768.1-n |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.365994989$ |
0.952738215 |
\( -\frac{827842288}{3} a + 770256176 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 128 a - 356\) , \( 1116 a - 3116\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(128a-356\right){x}+1116a-3116$ |
| 768.1-n2 |
768.1-n |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$4.365994989$ |
0.952738215 |
\( \frac{29376256}{3} a + \frac{52516864}{3} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 8 a - 21\) , \( 14 a - 40\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(8a-21\right){x}+14a-40$ |
| 768.1-o1 |
768.1-o |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{8} \cdot 3^{3} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$11.07281755$ |
2.416286884 |
\( -\frac{83200}{9} a + \frac{234752}{9} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( a + 2\) , \( 0\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(a+2\right){x}$ |
| 768.1-o2 |
768.1-o |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{6} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$11.07281755$ |
2.416286884 |
\( \frac{27376}{27} a + \frac{117536}{27} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -4 a - 8\) , \( -12 a - 20\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-4a-8\right){x}-12a-20$ |
| 768.1-p1 |
768.1-p |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$13.64169646$ |
2.976862221 |
\( -\frac{4864}{3} a + \frac{13568}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 1\) , \( -4 a - 7\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+{x}-4a-7$ |
| 768.1-p2 |
768.1-p |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{20} \cdot 3^{4} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$13.64169646$ |
2.976862221 |
\( -\frac{121322980}{9} a + \frac{112888388}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -45 a - 84\) , \( 135 a + 243\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-45a-84\right){x}+135a+243$ |
| 768.1-p3 |
768.1-p |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$13.64169646$ |
2.976862221 |
\( \frac{53200}{3} a + \frac{121184}{3} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -25 a - 44\) , \( -81 a - 145\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-25a-44\right){x}-81a-145$ |
| 768.1-p4 |
768.1-p |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{20} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$3.410424115$ |
2.976862221 |
\( \frac{5202690124}{3} a + \frac{9319516204}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -47 a + 128\) , \( -725 a + 2020\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-47a+128\right){x}-725a+2020$ |
| 768.1-q1 |
768.1-q |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{22} \cdot 3^{16} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$2.581159109$ |
$2.841754258$ |
3.201265131 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 16\) , \( 180\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+16{x}+180$ |
| 768.1-q2 |
768.1-q |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.290579554$ |
$11.36701703$ |
3.201265131 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+{x}$ |
| 768.1-q3 |
768.1-q |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.645289777$ |
$11.36701703$ |
3.201265131 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -4\) , \( -4\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-4{x}-4$ |
| 768.1-q4 |
768.1-q |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{20} \cdot 3^{8} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.290579554$ |
$11.36701703$ |
3.201265131 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -24\) , \( 36\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-24{x}+36$ |
| 768.1-q5 |
768.1-q |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{20} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.290579554$ |
$2.841754258$ |
3.201265131 |
\( \frac{28756228}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -64\) , \( -220\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-64{x}-220$ |
| 768.1-q6 |
768.1-q |
$6$ |
$8$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{22} \cdot 3^{4} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.645289777$ |
$11.36701703$ |
3.201265131 |
\( \frac{3065617154}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -384\) , \( 2772\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-384{x}+2772$ |
| 768.1-r1 |
768.1-r |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1.294261724$ |
$22.72318029$ |
3.208865982 |
\( -\frac{256}{3} a + \frac{5120}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( a + 2\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(a+2\right){x}$ |
| 768.1-r2 |
768.1-r |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{2} \) |
$2.15570$ |
$(-a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.647130862$ |
$22.72318029$ |
3.208865982 |
\( -\frac{77872}{3} a + 74480 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -4 a - 8\) , \( 4 a + 8\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-4a-8\right){x}+4a+8$ |
| 768.1-s1 |
768.1-s |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( - 2^{8} \cdot 3^{3} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1$ |
$26.16533078$ |
4.282307460 |
\( -\frac{83200}{9} a + \frac{234752}{9} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( a + 2\) , \( 0\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(a+2\right){x}$ |
| 768.1-s2 |
768.1-s |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
768.1 |
\( 2^{8} \cdot 3 \) |
\( 2^{16} \cdot 3^{6} \) |
$2.15570$ |
$(-a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$13.08266539$ |
4.282307460 |
\( \frac{27376}{27} a + \frac{117536}{27} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -4 a - 8\) , \( 12 a + 20\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-4a-8\right){x}+12a+20$ |
*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.
