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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 68544k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68544.w2 | 68544k1 | \([0, 0, 0, -67296, 6719400]\) | \(1219067475001344/4857223\) | \(134292501504\) | \([2]\) | \(168960\) | \(1.3470\) | \(\Gamma_0(N)\)-optimal |
68544.w1 | 68544k2 | \([0, 0, 0, -68316, 6505200]\) | \(79708988544624/4802079233\) | \(2124286186143744\) | \([2]\) | \(337920\) | \(1.6935\) |
Rank
sage: E.rank()
The elliptic curves in class 68544k have rank \(0\).
Complex multiplication
The elliptic curves in class 68544k do not have complex multiplication.Modular form 68544.2.a.k
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.