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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 4025.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4025.f1 | 4025c4 | \([1, -1, 0, -1052192, 415686091]\) | \(8244966675515989329/3081640625\) | \(48150634765625\) | \([2]\) | \(34560\) | \(1.9766\) | |
4025.f2 | 4025c3 | \([1, -1, 0, -137942, -10015159]\) | \(18577831198352049/7958740140575\) | \(124355314696484375\) | \([2]\) | \(34560\) | \(1.9766\) | |
4025.f3 | 4025c2 | \([1, -1, 0, -66067, 6444216]\) | \(2041085246738049/38897700625\) | \(607776572265625\) | \([2, 2]\) | \(17280\) | \(1.6301\) | |
4025.f4 | 4025c1 | \([1, -1, 0, 58, 294591]\) | \(1367631/2399636575\) | \(-37494321484375\) | \([2]\) | \(8640\) | \(1.2835\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4025.f have rank \(0\).
Complex multiplication
The elliptic curves in class 4025.f do not have complex multiplication.Modular form 4025.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.