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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 105966.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
105966.p1 | 105966p2 | \([1, -1, 0, -5721147, -4639750605]\) | \(1164800512406592125/150691848242418\) | \(2679237921865778466858\) | \([2]\) | \(6623232\) | \(2.8402\) | |
105966.p2 | 105966p1 | \([1, -1, 0, 545463, -379709127]\) | \(1009479798755875/4084868810988\) | \(-72627255899334836028\) | \([2]\) | \(3311616\) | \(2.4937\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 105966.p have rank \(1\).
Complex multiplication
The elliptic curves in class 105966.p do not have complex multiplication.Modular form 105966.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.