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SageMath
E = EllipticCurve("fp1")
E.isogeny_class()
Elliptic curves in class 259920.fp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259920.fp1 | 259920fp1 | \([0, 0, 0, -5187, -165566]\) | \(-14317849/2700\) | \(-2910438604800\) | \([]\) | \(497664\) | \(1.1157\) | \(\Gamma_0(N)\)-optimal |
259920.fp2 | 259920fp2 | \([0, 0, 0, 35853, 811186]\) | \(4728305591/3000000\) | \(-3233820672000000\) | \([]\) | \(1492992\) | \(1.6650\) |
Rank
sage: E.rank()
The elliptic curves in class 259920.fp have rank \(1\).
Complex multiplication
The elliptic curves in class 259920.fp do not have complex multiplication.Modular form 259920.2.a.fp
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.