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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 69360ck
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69360.t1 | 69360ck1 | \([0, -1, 0, -504401, -137731224]\) | \(-127157223424/16875\) | \(-1883454509070000\) | \([]\) | \(528768\) | \(1.9505\) | \(\Gamma_0(N)\)-optimal |
69360.t2 | 69360ck2 | \([0, -1, 0, 85159, -434928420]\) | \(611926016/732421875\) | \(-81747157511718750000\) | \([]\) | \(1586304\) | \(2.4998\) |
Rank
sage: E.rank()
The elliptic curves in class 69360ck have rank \(1\).
Complex multiplication
The elliptic curves in class 69360ck do not have complex multiplication.Modular form 69360.2.a.ck
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.