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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 150075.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
150075.f1 | 150075f1 | \([1, -1, 1, -9005, -324628]\) | \(7088952961/50025\) | \(569816015625\) | \([2]\) | \(393216\) | \(1.0884\) | \(\Gamma_0(N)\)-optimal |
150075.f2 | 150075f2 | \([1, -1, 1, -3380, -729628]\) | \(-374805361/20020005\) | \(-228040369453125\) | \([2]\) | \(786432\) | \(1.4350\) |
Rank
sage: E.rank()
The elliptic curves in class 150075.f have rank \(0\).
Complex multiplication
The elliptic curves in class 150075.f do not have complex multiplication.Modular form 150075.2.a.f
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.