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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 104742.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
104742.a1 | 104742bd3 | \([1, -1, 0, -47919564, -127666409796]\) | \(112763292123580561/1932612\) | \(208563936988297572\) | \([2]\) | \(9504000\) | \(2.8649\) | |
104742.a2 | 104742bd4 | \([1, -1, 0, -47871954, -127932787746]\) | \(-112427521449300721/466873642818\) | \(-50384145923853468402258\) | \([2]\) | \(19008000\) | \(3.2115\) | |
104742.a3 | 104742bd1 | \([1, -1, 0, -214344, 27895104]\) | \(10091699281/2737152\) | \(295388415913485312\) | \([2]\) | \(1900800\) | \(2.0602\) | \(\Gamma_0(N)\)-optimal |
104742.a4 | 104742bd2 | \([1, -1, 0, 547416, 181008864]\) | \(168105213359/228637728\) | \(-24674163616773319968\) | \([2]\) | \(3801600\) | \(2.4068\) |
Rank
sage: E.rank()
The elliptic curves in class 104742.a have rank \(0\).
Complex multiplication
The elliptic curves in class 104742.a do not have complex multiplication.Modular form 104742.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.