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 |
648.4-a1 |
648.4-a |
$1$ |
$1$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
648.4 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{18} \) |
$1.85890$ |
$(-a+2), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$1.208013344$ |
$1.131349575$ |
5.303518302 |
\( -\frac{80896}{9} a - \frac{388096}{27} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -261 a - 408\) , \( -3373 a - 5268\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-261a-408\right){x}-3373a-5268$ |
648.4-b1 |
648.4-b |
$2$ |
$2$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
648.4 |
\( 2^{3} \cdot 3^{4} \) |
\( - 2^{10} \cdot 3^{18} \) |
$1.85890$ |
$(-a+2), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.161399858$ |
1.494358965 |
\( -675 a + 1728 \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -2 a - 8\) , \( 141 a + 220\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a-8\right){x}+141a+220$ |
648.4-b2 |
648.4-b |
$2$ |
$2$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
648.4 |
\( 2^{3} \cdot 3^{4} \) |
\( - 2^{8} \cdot 3^{18} \) |
$1.85890$ |
$(-a+2), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.161399858$ |
1.494358965 |
\( 18495 a + 29916 \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -83 a + 208\) , \( -507 a + 1300\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-83a+208\right){x}-507a+1300$ |
648.4-c1 |
648.4-c |
$2$ |
$2$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
648.4 |
\( 2^{3} \cdot 3^{4} \) |
\( - 2^{10} \cdot 3^{6} \) |
$1.85890$ |
$(-a+2), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.451066374$ |
$8.577228616$ |
1.876691877 |
\( -675 a + 1728 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 0\) , \( -5 a - 8\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}-5a-8$ |
648.4-c2 |
648.4-c |
$2$ |
$2$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
648.4 |
\( 2^{3} \cdot 3^{4} \) |
\( - 2^{8} \cdot 3^{6} \) |
$1.85890$ |
$(-a+2), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.225533187$ |
$8.577228616$ |
1.876691877 |
\( 18495 a + 29916 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -9 a + 24\) , \( 14 a - 36\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-9a+24\right){x}+14a-36$ |
648.4-d1 |
648.4-d |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
648.4 |
\( 2^{3} \cdot 3^{4} \) |
\( - 2^{4} \cdot 3^{12} \) |
$1.85890$ |
$(-a+2), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.680814550$ |
$5.667089073$ |
1.871519699 |
\( -7659605 a + 19620476 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 39 a - 99\) , \( 205 a - 524\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(39a-99\right){x}+205a-524$ |
648.4-d2 |
648.4-d |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
648.4 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{11} \cdot 3^{12} \) |
$1.85890$ |
$(-a+2), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1.361629101$ |
$5.667089073$ |
1.871519699 |
\( 343 a + 686 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -51 a + 132\) , \( -948 a + 2428\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-51a+132\right){x}-948a+2428$ |
648.4-d3 |
648.4-d |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
648.4 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{10} \cdot 3^{12} \) |
$1.85890$ |
$(-a+2), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.680814550$ |
$11.33417814$ |
1.871519699 |
\( -1995 a + 7016 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 3 a - 12\) , \( -4 a + 12\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(3a-12\right){x}-4a+12$ |
648.4-d4 |
648.4-d |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
648.4 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{12} \) |
$1.85890$ |
$(-a+2), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.340407275$ |
$11.33417814$ |
1.871519699 |
\( 24225 a + 44228 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -123 a - 192\) , \( 1140 a + 1780\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-123a-192\right){x}+1140a+1780$ |
648.4-d5 |
648.4-d |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
648.4 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{11} \cdot 3^{12} \) |
$1.85890$ |
$(-a+2), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1.361629101$ |
$5.667089073$ |
1.871519699 |
\( -21069823 a + 54821634 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 48 a - 192\) , \( -427 a + 912\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(48a-192\right){x}-427a+912$ |
648.4-d6 |
648.4-d |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
648.4 |
\( 2^{3} \cdot 3^{4} \) |
\( - 2^{10} \cdot 3^{12} \) |
$1.85890$ |
$(-a+2), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.680814550$ |
$5.667089073$ |
1.871519699 |
\( 2701312025 a + 4218241728 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -717 a + 1836\) , \( 30184 a - 77316\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-717a+1836\right){x}+30184a-77316$ |
*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.