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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 435344.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
435344.z1 | 435344z4 | \([0, 0, 0, -334451, -74429966]\) | \(209267191953/55223\) | \(1091792377475072\) | \([2]\) | \(2457600\) | \(1.8700\) | |
435344.z2 | 435344z2 | \([0, 0, 0, -23491, -856830]\) | \(72511713/25921\) | \(512473973100544\) | \([2, 2]\) | \(1228800\) | \(1.5234\) | |
435344.z3 | 435344z1 | \([0, 0, 0, -9971, 373490]\) | \(5545233/161\) | \(3183068155904\) | \([2]\) | \(614400\) | \(1.1768\) | \(\Gamma_0(N)\)-optimal* |
435344.z4 | 435344z3 | \([0, 0, 0, 71149, -6024174]\) | \(2014698447/1958887\) | \(-38728390252883968\) | \([2]\) | \(2457600\) | \(1.8700\) |
Rank
sage: E.rank()
The elliptic curves in class 435344.z have rank \(0\).
Complex multiplication
The elliptic curves in class 435344.z do not have complex multiplication.Modular form 435344.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.