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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 48400cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
48400.t2 | 48400cm1 | \([0, 1, 0, 144192, 8046388]\) | \(24167/16\) | \(-219503494144000000\) | \([]\) | \(456192\) | \(2.0163\) | \(\Gamma_0(N)\)-optimal |
48400.t1 | 48400cm2 | \([0, 1, 0, -2517808, 1578626388]\) | \(-128667913/4096\) | \(-56192894500864000000\) | \([]\) | \(1368576\) | \(2.5656\) |
Rank
sage: E.rank()
The elliptic curves in class 48400cm have rank \(1\).
Complex multiplication
The elliptic curves in class 48400cm do not have complex multiplication.Modular form 48400.2.a.cm
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.