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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 72450.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.a1 | 72450e2 | \([1, -1, 0, -2512392, -1532149984]\) | \(-5702623460245179/252448\) | \(-77639593500000\) | \([]\) | \(1710720\) | \(2.1474\) | |
72450.a2 | 72450e1 | \([1, -1, 0, -28392, -2465984]\) | \(-5999796014211/2790817792\) | \(-1177376256000000\) | \([]\) | \(570240\) | \(1.5980\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.a have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.a do not have complex multiplication.Modular form 72450.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.