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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 72930.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72930.cp1 | 72930cr8 | \([1, 0, 0, -627818501, -6054844853445]\) | \(27366927465515907102849247581649/820749800755371093750\) | \(820749800755371093750\) | \([2]\) | \(25214976\) | \(3.5195\) | |
| 72930.cp2 | 72930cr6 | \([1, 0, 0, -39290231, -94348240539]\) | \(6707759134413896021566057969/36584858534733500062500\) | \(36584858534733500062500\) | \([2, 2]\) | \(12607488\) | \(3.1729\) | |
| 72930.cp3 | 72930cr7 | \([1, 0, 0, -17661481, -197720688289]\) | \(-609263370915155282374717969/16535544894232646154765750\) | \(-16535544894232646154765750\) | \([2]\) | \(25214976\) | \(3.5195\) | |
| 72930.cp4 | 72930cr5 | \([1, 0, 0, -8330411, -6992382759]\) | \(63932693528384163534877489/15880324290759886815000\) | \(15880324290759886815000\) | \([6]\) | \(8404992\) | \(2.9702\) | |
| 72930.cp5 | 72930cr3 | \([1, 0, 0, -3859011, 401927985]\) | \(6355529936351173360371889/3608069075870283882000\) | \(3608069075870283882000\) | \([2]\) | \(6303744\) | \(2.8264\) | |
| 72930.cp6 | 72930cr2 | \([1, 0, 0, -2877491, 1787909025]\) | \(2634909136670192647897009/143521963154656257600\) | \(143521963154656257600\) | \([2, 6]\) | \(4202496\) | \(2.6236\) | |
| 72930.cp7 | 72930cr1 | \([1, 0, 0, -2838771, 1840715361]\) | \(2529966680458237378423729/8164079019233280\) | \(8164079019233280\) | \([6]\) | \(2101248\) | \(2.2771\) | \(\Gamma_0(N)\)-optimal |
| 72930.cp8 | 72930cr4 | \([1, 0, 0, 1955909, 7188750185]\) | \(827503107744014175728591/22802818589846166694680\) | \(-22802818589846166694680\) | \([6]\) | \(8404992\) | \(2.9702\) |
Rank
sage: E.rank()
The elliptic curves in class 72930.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 72930.cp do not have complex multiplication.Modular form 72930.2.a.cp
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
