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SageMath
E = EllipticCurve("ii1")
E.isogeny_class()
Elliptic curves in class 270400.ii
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
270400.ii1 | 270400ii1 | \([0, -1, 0, -227412033, -1319726704063]\) | \(65787589563409/10400000\) | \(205614340505600000000000\) | \([2]\) | \(61931520\) | \(3.4839\) | \(\Gamma_0(N)\)-optimal |
270400.ii2 | 270400ii2 | \([0, -1, 0, -205780033, -1580889840063]\) | \(-48743122863889/26406250000\) | \(-522067661440000000000000000\) | \([2]\) | \(123863040\) | \(3.8304\) |
Rank
sage: E.rank()
The elliptic curves in class 270400.ii have rank \(1\).
Complex multiplication
The elliptic curves in class 270400.ii do not have complex multiplication.Modular form 270400.2.a.ii
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.