| 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 |
| 282.1-a1 |
282.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( - 2 \cdot 3^{2} \cdot 47^{8} \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.526057320$ |
2.429754685 |
\( -\frac{119826757520628220073}{142867719970566} a + \frac{192068797874750696711}{142867719970566} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 2464 a - 4293\) , \( 86766 a - 150351\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(2464a-4293\right){x}+86766a-150351$ |
| 282.1-a2 |
282.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( - 2^{2} \cdot 3^{16} \cdot 47 \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$4.208458565$ |
2.429754685 |
\( -\frac{309636401003783}{616734} a + \frac{268152986824457}{308367} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 1269 a - 2198\) , \( -33030 a + 57207\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(1269a-2198\right){x}-33030a+57207$ |
| 282.1-a3 |
282.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( - 2^{8} \cdot 3^{4} \cdot 47 \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$16.83383426$ |
2.429754685 |
\( \frac{462281}{846} a + \frac{6407911}{6768} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -a + 2\) , \( -24 a + 41\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-a+2\right){x}-24a+41$ |
| 282.1-a4 |
282.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( 2^{4} \cdot 3^{8} \cdot 47^{2} \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$8.416917130$ |
2.429754685 |
\( \frac{3479736953}{178929} a + \frac{54492051847}{715716} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 79 a - 138\) , \( -540 a + 933\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(79a-138\right){x}-540a+933$ |
| 282.1-a5 |
282.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( 2^{2} \cdot 3^{4} \cdot 47^{4} \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$1$ |
$2.104229282$ |
2.429754685 |
\( \frac{301663123443422609}{87834258} a + \frac{261248039917326193}{43917129} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 169 a - 318\) , \( 846 a - 1533\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(169a-318\right){x}+846a-1533$ |
| 282.1-a6 |
282.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( - 2 \cdot 3^{2} \cdot 47^{2} \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.526057320$ |
2.429754685 |
\( \frac{541569619576386201732041}{13254} a + \frac{938026096942049393503993}{13254} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -686 a + 777\) , \( 6150 a - 14139\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-686a+777\right){x}+6150a-14139$ |
| 282.1-b1 |
282.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( 2^{2} \cdot 3^{7} \cdot 47 \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$3.103808503$ |
1.791984674 |
\( -\frac{31900747}{3807} a - \frac{38947367}{2538} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( a - 4\) , \( 3 a - 6\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(a-4\right){x}+3a-6$ |
| 282.1-c1 |
282.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( 2^{2} \cdot 3 \cdot 47 \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$4.410111112$ |
2.546178837 |
\( \frac{758739929}{141} a - \frac{876017351}{94} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 10 a + 17\) , \( -4 a - 7\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(10a+17\right){x}-4a-7$ |
| 282.1-d1 |
282.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( 2^{2} \cdot 3 \cdot 47 \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.038199064$ |
$28.15616714$ |
1.241925736 |
\( \frac{758739929}{141} a - \frac{876017351}{94} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 8 a + 16\) , \( 13 a + 23\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(8a+16\right){x}+13a+23$ |
| 282.1-e1 |
282.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( 2^{2} \cdot 3^{7} \cdot 47 \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.008322890$ |
$23.64481833$ |
1.590660710 |
\( -\frac{31900747}{3807} a - \frac{38947367}{2538} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( a - 4\) , \( -3 a + 5\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-4\right){x}-3a+5$ |
| 282.1-f1 |
282.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( - 2 \cdot 3^{2} \cdot 47^{8} \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.478439175$ |
$4.588801779$ |
1.267550887 |
\( -\frac{119826757520628220073}{142867719970566} a + \frac{192068797874750696711}{142867719970566} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 2464 a - 4293\) , \( -86766 a + 150350\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2464a-4293\right){x}-86766a+150350$ |
| 282.1-f2 |
282.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( - 2^{2} \cdot 3^{16} \cdot 47 \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.956878350$ |
$2.294400889$ |
1.267550887 |
\( -\frac{309636401003783}{616734} a + \frac{268152986824457}{308367} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 1269 a - 2198\) , \( 33030 a - 57208\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1269a-2198\right){x}+33030a-57208$ |
| 282.1-f3 |
282.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( - 2^{8} \cdot 3^{4} \cdot 47 \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.239219587$ |
$9.177603559$ |
1.267550887 |
\( \frac{462281}{846} a + \frac{6407911}{6768} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -a + 2\) , \( 24 a - 42\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a+2\right){x}+24a-42$ |
| 282.1-f4 |
282.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( 2^{4} \cdot 3^{8} \cdot 47^{2} \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.478439175$ |
$9.177603559$ |
1.267550887 |
\( \frac{3479736953}{178929} a + \frac{54492051847}{715716} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 79 a - 138\) , \( 540 a - 934\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(79a-138\right){x}+540a-934$ |
| 282.1-f5 |
282.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( 2^{2} \cdot 3^{4} \cdot 47^{4} \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.956878350$ |
$9.177603559$ |
1.267550887 |
\( \frac{301663123443422609}{87834258} a + \frac{261248039917326193}{43917129} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 169 a - 318\) , \( -846 a + 1532\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(169a-318\right){x}-846a+1532$ |
| 282.1-f6 |
282.1-f |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
282.1 |
\( 2 \cdot 3 \cdot 47 \) |
\( - 2 \cdot 3^{2} \cdot 47^{2} \) |
$1.26850$ |
$(a+1), (a), (4a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.913756700$ |
$4.588801779$ |
1.267550887 |
\( \frac{541569619576386201732041}{13254} a + \frac{938026096942049393503993}{13254} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -686 a + 777\) , \( -6150 a + 14138\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-686a+777\right){x}-6150a+14138$ |
*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.
