| 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 |
| 147.1-a1 |
147.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{16} \) |
$1.78741$ |
$(-2a+7), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.814020435$ |
0.566811077 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -34\) , \( -217\bigr] \) |
${y}^2+{x}{y}={x}^{3}-34{x}-217$ |
| 147.1-a2 |
147.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{2} \) |
$1.78741$ |
$(-2a+7), (7)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$13.02432697$ |
0.566811077 |
\( \frac{103823}{63} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}$ |
| 147.1-a3 |
147.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{8} \cdot 7^{4} \) |
$1.78741$ |
$(-2a+7), (7)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$13.02432697$ |
0.566811077 |
\( \frac{7189057}{3969} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -4\) , \( -1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-4{x}-1$ |
| 147.1-a4 |
147.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{16} \cdot 7^{2} \) |
$1.78741$ |
$(-2a+7), (7)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$13.02432697$ |
0.566811077 |
\( \frac{6570725617}{45927} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -39\) , \( 90\bigr] \) |
${y}^2+{x}{y}={x}^{3}-39{x}+90$ |
| 147.1-a5 |
147.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{8} \) |
$1.78741$ |
$(-2a+7), (7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$3.256081743$ |
0.566811077 |
\( \frac{13027640977}{21609} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -49\) , \( -136\bigr] \) |
${y}^2+{x}{y}={x}^{3}-49{x}-136$ |
| 147.1-a6 |
147.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{4} \) |
$1.78741$ |
$(-2a+7), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$0.814020435$ |
0.566811077 |
\( \frac{53297461115137}{147} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -784\) , \( -8515\bigr] \) |
${y}^2+{x}{y}={x}^{3}-784{x}-8515$ |
| 147.1-b1 |
147.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{16} \) |
$1.78741$ |
$(-2a+7), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.651881942$ |
2.542844193 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -12519 a - 29697\) , \( 3604391 a + 8550628\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-12519a-29697\right){x}+3604391a+8550628$ |
| 147.1-b2 |
147.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{2} \) |
$1.78741$ |
$(-2a+7), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$14.60752776$ |
2.542844193 |
\( \frac{103823}{63} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( 361 a + 858\) , \( 1946 a + 4615\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(361a+858\right){x}+1946a+4615$ |
| 147.1-b3 |
147.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{8} \cdot 7^{4} \) |
$1.78741$ |
$(-2a+7), (7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$14.60752776$ |
2.542844193 |
\( \frac{7189057}{3969} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 1480 a - 4987\) , \( -10461 a + 35272\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1480a-4987\right){x}-10461a+35272$ |
| 147.1-b4 |
147.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{16} \cdot 7^{2} \) |
$1.78741$ |
$(-2a+7), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.651881942$ |
2.542844193 |
\( \frac{6570725617}{45927} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -14359 a - 34062\) , \( -1599934 a - 3795495\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-14359a-34062\right){x}-1599934a-3795495$ |
| 147.1-b5 |
147.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{8} \) |
$1.78741$ |
$(-2a+7), (7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$14.60752776$ |
2.542844193 |
\( \frac{13027640977}{21609} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -18039 a - 42792\) , \( 2204216 a + 5229019\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-18039a-42792\right){x}+2204216a+5229019$ |
| 147.1-b6 |
147.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
147.1 |
\( 3 \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{4} \) |
$1.78741$ |
$(-2a+7), (7)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$14.60752776$ |
2.542844193 |
\( \frac{53297461115137}{147} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -288519 a - 684447\) , \( 142523801 a + 338106550\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-288519a-684447\right){x}+142523801a+338106550$ |
*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.
