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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 70070.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 70070.z1 | 70070w4 | \([1, 1, 0, -3870311967, 92650522961669]\) | \(54497099771831721530744218729/16209843781074944000000\) | \(1907071910999686086656000000\) | \([2]\) | \(69672960\) | \(4.2130\) | |
| 70070.z2 | 70070w3 | \([1, 1, 0, -273696287, 1042563622661]\) | \(19272683606216463573689449/7161126378530668544000\) | \(842499357307754623533056000\) | \([2]\) | \(34836480\) | \(3.8665\) | |
| 70070.z3 | 70070w2 | \([1, 1, 0, -129024032, -399631805824]\) | \(2019051077229077416165369/582160888682835862400\) | \(68490646392646956375497600\) | \([2]\) | \(23224320\) | \(3.6637\) | |
| 70070.z4 | 70070w1 | \([1, 1, 0, -118267552, -495035329536]\) | \(1555006827939811751684089/221961497899581440\) | \(26113548266387856834560\) | \([2]\) | \(11612160\) | \(3.3172\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 70070.z have rank \(0\).
Complex multiplication
The elliptic curves in class 70070.z do not have complex multiplication.Modular form 70070.2.a.z
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
