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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 116886.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
116886.p1 | 116886w4 | \([1, 0, 1, -21640732, 38746796900]\) | \(632678989847546725777/80515134\) | \(142637471304174\) | \([2]\) | \(4915200\) | \(2.5759\) | |
116886.p2 | 116886w3 | \([1, 0, 1, -1547472, 419454484]\) | \(231331938231569617/90942310746882\) | \(161109850969057022802\) | \([2]\) | \(4915200\) | \(2.5759\) | |
116886.p3 | 116886w2 | \([1, 0, 1, -1352662, 605225300]\) | \(154502321244119857/55101928644\) | \(97616427810493284\) | \([2, 2]\) | \(2457600\) | \(2.2294\) | |
116886.p4 | 116886w1 | \([1, 0, 1, -72482, 12245924]\) | \(-23771111713777/22848457968\) | \(-40477437046248048\) | \([2]\) | \(1228800\) | \(1.8828\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 116886.p have rank \(1\).
Complex multiplication
The elliptic curves in class 116886.p do not have complex multiplication.Modular form 116886.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.