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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 44100cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
44100.o2 | 44100cf1 | \([0, 0, 0, 735, -22295]\) | \(1280/7\) | \(-240145138800\) | \([]\) | \(41472\) | \(0.86578\) | \(\Gamma_0(N)\)-optimal |
44100.o1 | 44100cf2 | \([0, 0, 0, -43365, -3479735]\) | \(-262885120/343\) | \(-11767111801200\) | \([]\) | \(124416\) | \(1.4151\) |
Rank
sage: E.rank()
The elliptic curves in class 44100cf have rank \(2\).
Complex multiplication
The elliptic curves in class 44100cf do not have complex multiplication.Modular form 44100.2.a.cf
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.