| 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 |
| 199.2-a1 |
199.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
199.2 |
\( 199 \) |
\( -199 \) |
$0.75048$ |
$(3a+13)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1[2] |
$1$ |
\( 1 \) |
$1$ |
$48.28902046$ |
0.863820258 |
\( -\frac{537477120}{199} a - \frac{280571904}{199} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -4 \phi - 5\) , \( 6 \phi + 5\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-4\phi-5\right){x}+6\phi+5$ |
| 199.2-a2 |
199.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
199.2 |
\( 199 \) |
\( - 199^{5} \) |
$0.75048$ |
$(3a+13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.4[2] |
$1$ |
\( 1 \) |
$1$ |
$1.931560818$ |
0.863820258 |
\( -\frac{1054854363044265984}{312079600999} a + \frac{1707328883873968128}{312079600999} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 76 \phi - 55\) , \( 174 \phi - 289\bigr] \) |
${y}^2+{y}={x}^{3}+\left(76\phi-55\right){x}+174\phi-289$ |
| 199.2-b1 |
199.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
199.2 |
\( 199 \) |
\( -199 \) |
$0.75048$ |
$(3a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$8.322257936$ |
0.930456723 |
\( -\frac{23123880}{199} a + \frac{35419707}{199} \) |
\( \bigl[1\) , \( -1\) , \( \phi\) , \( -2\) , \( -1\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}-{x}^{2}-2{x}-1$ |
| 199.2-b2 |
199.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
199.2 |
\( 199 \) |
\( - 199^{2} \) |
$0.75048$ |
$(3a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$4.161128968$ |
0.930456723 |
\( \frac{2048009135562}{39601} a + \frac{1265736486645}{39601} \) |
\( \bigl[\phi + 1\) , \( 1\) , \( 1\) , \( 3 \phi - 7\) , \( 23 \phi - 42\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(3\phi-7\right){x}+23\phi-42$ |
| 199.2-c1 |
199.2-c |
$3$ |
$9$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
199.2 |
\( 199 \) |
\( -199 \) |
$0.75048$ |
$(3a+13)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1.388777116$ |
$0.529573384$ |
0.657814883 |
\( -\frac{376268319166603722752}{199} a + \frac{608814929301329936384}{199} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -392 \phi - 441\) , \( -5756 \phi - 4705\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(-392\phi-441\right){x}-5756\phi-4705$ |
| 199.2-c2 |
199.2-c |
$3$ |
$9$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
199.2 |
\( 199 \) |
\( -199 \) |
$0.75048$ |
$(3a+13)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$0.154308568$ |
$42.89544410$ |
0.657814883 |
\( -\frac{524288}{199} a + \frac{622592}{199} \) |
\( \bigl[0\) , \( -\phi - 1\) , \( 1\) , \( \phi\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\phi{x}$ |
| 199.2-c3 |
199.2-c |
$3$ |
$9$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
199.2 |
\( 199 \) |
\( - 199^{3} \) |
$0.75048$ |
$(3a+13)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.462925705$ |
$4.766160456$ |
0.657814883 |
\( -\frac{5698721480704}{7880599} a + \frac{9226784440320}{7880599} \) |
\( \bigl[0\) , \( -\phi - 1\) , \( 1\) , \( 11 \phi - 10\) , \( 6 \phi - 21\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(11\phi-10\right){x}+6\phi-21$ |
*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.
