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SageMath
E = EllipticCurve("hp1")
E.isogeny_class()
Elliptic curves in class 139650hp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
139650.k2 | 139650hp1 | \([1, 1, 0, -193575, -34252875]\) | \(-3491055413/175104\) | \(-40235958000000000\) | \([2]\) | \(1843200\) | \(1.9470\) | \(\Gamma_0(N)\)-optimal |
139650.k1 | 139650hp2 | \([1, 1, 0, -3133575, -2136352875]\) | \(14809006736693/34656\) | \(7963366687500000\) | \([2]\) | \(3686400\) | \(2.2936\) |
Rank
sage: E.rank()
The elliptic curves in class 139650hp have rank \(0\).
Complex multiplication
The elliptic curves in class 139650hp do not have complex multiplication.Modular form 139650.2.a.hp
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.