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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 10010e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10010.g1 | 10010e1 | \([1, 0, 1, -39, 1482]\) | \(-6321363049/949588640\) | \(-949588640\) | \([3]\) | \(4320\) | \(0.40201\) | \(\Gamma_0(N)\)-optimal |
10010.g2 | 10010e2 | \([1, 0, 1, 346, -39944]\) | \(4599970967591/692916224000\) | \(-692916224000\) | \([]\) | \(12960\) | \(0.95132\) |
Rank
sage: E.rank()
The elliptic curves in class 10010e have rank \(0\).
Complex multiplication
The elliptic curves in class 10010e do not have complex multiplication.Modular form 10010.2.a.e
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.