| 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 |
| 20.1-a1 |
20.1-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{36} \cdot 5^{4} \) |
$14.47032$ |
$(1/2a^3-1/2a^2-7/2a+5/2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$189.1172742$ |
1.698323258 |
\( -\frac{2062826524589}{25600} a^{3} + \frac{4002071809497}{25600} a^{2} + \frac{16989585953303}{25600} a - \frac{32961272479261}{25600} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{7}{2} a\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{9}{2} a + \frac{9}{2}\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a\) , \( -\frac{25}{2} a^{3} - 31 a^{2} + \frac{213}{2} a + 227\) , \( -100 a^{3} - \frac{385}{2} a^{2} + 824 a + \frac{3181}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{7}{2}a\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{9}{2}a+\frac{9}{2}\right){x}^{2}+\left(-\frac{25}{2}a^{3}-31a^{2}+\frac{213}{2}a+227\right){x}-100a^{3}-\frac{385}{2}a^{2}+824a+\frac{3181}{2}$ |
| 20.1-a2 |
20.1-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{12} \cdot 5^{12} \) |
$14.47032$ |
$(1/2a^3-1/2a^2-7/2a+5/2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$189.1172742$ |
1.698323258 |
\( -\frac{2028817931053}{250000} a^{3} + \frac{3032952547803}{125000} a^{2} + \frac{7629088540237}{250000} a - \frac{1142292115229}{12500} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{5}{2} a + 1\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{7}{2}\) , \( 0\) , \( 57 a^{3} - \frac{143}{2} a^{2} - 287 a + \frac{247}{2}\) , \( -597 a^{3} + \frac{2261}{2} a^{2} + 2805 a - \frac{6329}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{5}{2}a+1\right){x}{y}={x}^{3}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{7}{2}\right){x}^{2}+\left(57a^{3}-\frac{143}{2}a^{2}-287a+\frac{247}{2}\right){x}-597a^{3}+\frac{2261}{2}a^{2}+2805a-\frac{6329}{2}$ |
| 20.1-a3 |
20.1-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{6} \cdot 5^{24} \) |
$14.47032$ |
$(1/2a^3-1/2a^2-7/2a+5/2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$94.55863714$ |
1.698323258 |
\( -\frac{1455229050566292019}{1953125000} a^{3} - \frac{2823809491238563383}{1953125000} a^{2} + \frac{11985361149037390153}{1953125000} a + \frac{23257104737525574349}{1953125000} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{7}{2} a - \frac{5}{2}\) , \( -\frac{1}{2} a^{2} + a + \frac{9}{2}\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{5}{2}\) , \( \frac{611}{2} a^{3} - 695 a^{2} - \frac{2631}{2} a + 2280\) , \( -8588 a^{3} + \frac{46545}{2} a^{2} + 33599 a - \frac{170223}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{7}{2}a-\frac{5}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{5}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{2}+a+\frac{9}{2}\right){x}^{2}+\left(\frac{611}{2}a^{3}-695a^{2}-\frac{2631}{2}a+2280\right){x}-8588a^{3}+\frac{46545}{2}a^{2}+33599a-\frac{170223}{2}$ |
| 20.1-a4 |
20.1-a |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{18} \cdot 5^{8} \) |
$14.47032$ |
$(1/2a^3-1/2a^2-7/2a+5/2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$94.55863714$ |
1.698323258 |
\( \frac{4506837263097866069597}{40000} a^{3} - \frac{8743649944588441599153}{40000} a^{2} - \frac{7423723612473221373661}{8000} a + \frac{72013295315208218404901}{40000} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{7}{2} a\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{9}{2} a + \frac{9}{2}\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a\) , \( \frac{135}{2} a^{3} + 49 a^{2} - \frac{1067}{2} a - 733\) , \( -172 a^{3} - \frac{1681}{2} a^{2} + 2192 a + \frac{12317}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{7}{2}a\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{9}{2}a+\frac{9}{2}\right){x}^{2}+\left(\frac{135}{2}a^{3}+49a^{2}-\frac{1067}{2}a-733\right){x}-172a^{3}-\frac{1681}{2}a^{2}+2192a+\frac{12317}{2}$ |
| 20.1-b1 |
20.1-b |
$2$ |
$3$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{2} \cdot 5^{3} \) |
$14.47032$ |
$(1/2a^3-1/2a^2-7/2a+5/2), (-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.038685298$ |
$778.1746306$ |
3.244093983 |
\( -\frac{45736318}{25} a^{3} + \frac{177288981}{50} a^{2} + \frac{753945911}{50} a - \frac{730980408}{25} \) |
\( \bigl[a\) , \( -\frac{1}{2} a^{3} + \frac{9}{2} a - 1\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{5}{2}\) , \( -4 a^{2} + 24\) , \( -\frac{1}{2} a^{3} - \frac{3}{2} a^{2} + \frac{7}{2} a + \frac{9}{2}\bigr] \) |
${y}^2+a{x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{5}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{9}{2}a-1\right){x}^{2}+\left(-4a^{2}+24\right){x}-\frac{1}{2}a^{3}-\frac{3}{2}a^{2}+\frac{7}{2}a+\frac{9}{2}$ |
| 20.1-b2 |
20.1-b |
$2$ |
$3$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{6} \cdot 5 \) |
$14.47032$ |
$(1/2a^3-1/2a^2-7/2a+5/2), (-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.116055896$ |
$778.1746306$ |
3.244093983 |
\( -\frac{2135237909}{40} a^{3} - \frac{4142533731}{40} a^{2} + \frac{17585965949}{40} a + \frac{34118238951}{40} \) |
\( \bigl[\frac{1}{2} a^{2} + a - \frac{7}{2}\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{7}{2} a + \frac{7}{2}\) , \( \frac{1}{2} a^{2} - \frac{5}{2}\) , \( -a^{3} + \frac{9}{2} a^{2} - a - \frac{15}{2}\) , \( \frac{13}{2} a^{3} - 17 a^{2} - \frac{61}{2} a + 76\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}-\frac{5}{2}\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{7}{2}a+\frac{7}{2}\right){x}^{2}+\left(-a^{3}+\frac{9}{2}a^{2}-a-\frac{15}{2}\right){x}+\frac{13}{2}a^{3}-17a^{2}-\frac{61}{2}a+76$ |
| 20.1-c1 |
20.1-c |
$2$ |
$3$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{2} \cdot 5^{3} \) |
$14.47032$ |
$(1/2a^3-1/2a^2-7/2a+5/2), (-a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$129.2017712$ |
1.160266157 |
\( -\frac{45736318}{25} a^{3} + \frac{177288981}{50} a^{2} + \frac{753945911}{50} a - \frac{730980408}{25} \) |
\( \bigl[\frac{1}{2} a^{2} - \frac{5}{2}\) , \( -\frac{1}{2} a^{3} + \frac{5}{2} a\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{7}{2} a - \frac{5}{2}\) , \( \frac{3}{2} a^{2} - a - \frac{15}{2}\) , \( \frac{1}{2} a^{2} - a - \frac{11}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}-\frac{5}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{7}{2}a-\frac{5}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a\right){x}^{2}+\left(\frac{3}{2}a^{2}-a-\frac{15}{2}\right){x}+\frac{1}{2}a^{2}-a-\frac{11}{2}$ |
| 20.1-c2 |
20.1-c |
$2$ |
$3$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{6} \cdot 5 \) |
$14.47032$ |
$(1/2a^3-1/2a^2-7/2a+5/2), (-a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$129.2017712$ |
1.160266157 |
\( -\frac{2135237909}{40} a^{3} - \frac{4142533731}{40} a^{2} + \frac{17585965949}{40} a + \frac{34118238951}{40} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{7}{2}\) , \( a\) , \( \frac{1}{2} a^{2} - \frac{7}{2}\) , \( -\frac{9}{2} a^{3} + 19 a^{2} + \frac{35}{2} a - 70\) , \( 17 a^{3} - \frac{83}{2} a^{2} - 63 a + \frac{311}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}-\frac{7}{2}\right){y}={x}^{3}+a{x}^{2}+\left(-\frac{9}{2}a^{3}+19a^{2}+\frac{35}{2}a-70\right){x}+17a^{3}-\frac{83}{2}a^{2}-63a+\frac{311}{2}$ |
| 20.1-d1 |
20.1-d |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{36} \cdot 5^{4} \) |
$14.47032$ |
$(1/2a^3-1/2a^2-7/2a+5/2), (-a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.511945060$ |
$22.84119804$ |
2.481041532 |
\( -\frac{2062826524589}{25600} a^{3} + \frac{4002071809497}{25600} a^{2} + \frac{16989585953303}{25600} a - \frac{32961272479261}{25600} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + \frac{5}{2} a - \frac{5}{2}\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( -30 a^{3} - 58 a^{2} + 246 a + 475\) , \( 245 a^{3} + 473 a^{2} - 2013 a - 3901\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+\frac{5}{2}a-\frac{5}{2}\right){x}^{2}+\left(-30a^{3}-58a^{2}+246a+475\right){x}+245a^{3}+473a^{2}-2013a-3901$ |
| 20.1-d2 |
20.1-d |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{12} \cdot 5^{12} \) |
$14.47032$ |
$(1/2a^3-1/2a^2-7/2a+5/2), (-a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.503981686$ |
$22.84119804$ |
2.481041532 |
\( -\frac{2028817931053}{250000} a^{3} + \frac{3032952547803}{125000} a^{2} + \frac{7629088540237}{250000} a - \frac{1142292115229}{12500} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{7}{2} a + \frac{7}{2}\) , \( \frac{1}{2} a^{2} + a - \frac{7}{2}\) , \( 109 a^{3} - 281 a^{2} - 447 a + 1019\) , \( -\frac{4051}{2} a^{3} + 5934 a^{2} + \frac{15015}{2} a - 22545\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+1\right){x}{y}+\left(\frac{1}{2}a^{2}+a-\frac{7}{2}\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{7}{2}a+\frac{7}{2}\right){x}^{2}+\left(109a^{3}-281a^{2}-447a+1019\right){x}-\frac{4051}{2}a^{3}+5934a^{2}+\frac{15015}{2}a-22545$ |
| 20.1-d3 |
20.1-d |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{6} \cdot 5^{24} \) |
$14.47032$ |
$(1/2a^3-1/2a^2-7/2a+5/2), (-a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1.007963373$ |
$11.42059902$ |
2.481041532 |
\( -\frac{1455229050566292019}{1953125000} a^{3} - \frac{2823809491238563383}{1953125000} a^{2} + \frac{11985361149037390153}{1953125000} a + \frac{23257104737525574349}{1953125000} \) |
\( \bigl[\frac{1}{2} a^{2} + a - \frac{5}{2}\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{5}{2} a + \frac{5}{2}\) , \( \frac{1}{2} a^{2} + a - \frac{7}{2}\) , \( \frac{1435}{2} a^{3} - 1985 a^{2} - \frac{5531}{2} a + 7339\) , \( -\frac{71031}{2} a^{3} + \frac{204731}{2} a^{2} + \frac{266547}{2} a - \frac{772191}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+a-\frac{5}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}+a-\frac{7}{2}\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{5}{2}a+\frac{5}{2}\right){x}^{2}+\left(\frac{1435}{2}a^{3}-1985a^{2}-\frac{5531}{2}a+7339\right){x}-\frac{71031}{2}a^{3}+\frac{204731}{2}a^{2}+\frac{266547}{2}a-\frac{772191}{2}$ |
| 20.1-d4 |
20.1-d |
$4$ |
$6$ |
4.4.12400.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{18} \cdot 5^{8} \) |
$14.47032$ |
$(1/2a^3-1/2a^2-7/2a+5/2), (-a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$3.023890121$ |
$11.42059902$ |
2.481041532 |
\( \frac{4506837263097866069597}{40000} a^{3} - \frac{8743649944588441599153}{40000} a^{2} - \frac{7423723612473221373661}{8000} a + \frac{72013295315208218404901}{40000} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + \frac{5}{2} a - \frac{5}{2}\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( 170 a^{3} + 302 a^{2} - 1394 a - 2605\) , \( 2485 a^{3} + 4649 a^{2} - 20205 a - 38557\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+\frac{5}{2}a-\frac{5}{2}\right){x}^{2}+\left(170a^{3}+302a^{2}-1394a-2605\right){x}+2485a^{3}+4649a^{2}-20205a-38557$ |
*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.
