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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 4275c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4275.g2 | 4275c1 | \([1, -1, 1, -21305, -1204928]\) | \(-27818127/361\) | \(-13878052734375\) | \([2]\) | \(7680\) | \(1.3306\) | \(\Gamma_0(N)\)-optimal |
4275.g1 | 4275c2 | \([1, -1, 1, -341930, -76872428]\) | \(115003963647/19\) | \(730423828125\) | \([2]\) | \(15360\) | \(1.6772\) |
Rank
sage: E.rank()
The elliptic curves in class 4275c have rank \(1\).
Complex multiplication
The elliptic curves in class 4275c do not have complex multiplication.Modular form 4275.2.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 Cremona 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 Cremona labels.