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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 106722k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
106722.dm2 | 106722k1 | \([1, -1, 0, 7782, 18073908]\) | \(189/512\) | \(-141180318135166464\) | \([]\) | \(1360800\) | \(1.9697\) | \(\Gamma_0(N)\)-optimal |
106722.dm1 | 106722k2 | \([1, -1, 0, -4972578, 4269641228]\) | \(-67645179/8\) | \(-1608132061258380504\) | \([]\) | \(4082400\) | \(2.5190\) |
Rank
sage: E.rank()
The elliptic curves in class 106722k have rank \(0\).
Complex multiplication
The elliptic curves in class 106722k do not have complex multiplication.Modular form 106722.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.