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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 4140.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4140.c1 | 4140a2 | \([0, 0, 0, -10023, 386222]\) | \(16110654114672/330625\) | \(2285280000\) | \([2]\) | \(3840\) | \(0.91519\) | |
4140.c2 | 4140a1 | \([0, 0, 0, -648, 5597]\) | \(69657034752/8984375\) | \(3881250000\) | \([2]\) | \(1920\) | \(0.56861\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4140.c have rank \(1\).
Complex multiplication
The elliptic curves in class 4140.c do not have complex multiplication.Modular form 4140.2.a.c
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.