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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 112710.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
112710.a1 | 112710h2 | \([1, 1, 0, -21298, -1089548]\) | \(217482980991353/23168683200\) | \(113827740561600\) | \([2]\) | \(700416\) | \(1.4316\) | |
112710.a2 | 112710h1 | \([1, 1, 0, -4978, 114868]\) | \(2777652643193/404951040\) | \(1989524459520\) | \([2]\) | \(350208\) | \(1.0850\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 112710.a have rank \(2\).
Complex multiplication
The elliptic curves in class 112710.a do not have complex multiplication.Modular form 112710.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.