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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 348726.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
348726.b1 | 348726b3 | \([1, 1, 0, -64564496, 199655028570]\) | \(632678989847546725777/80515134\) | \(3787905412863054\) | \([2]\) | \(27648000\) | \(2.8492\) | |
348726.b2 | 348726b4 | \([1, 1, 0, -4616836, 2160302326]\) | \(231331938231569617/90942310746882\) | \(4278461129262831693042\) | \([2]\) | \(27648000\) | \(2.8492\) | |
348726.b3 | 348726b2 | \([1, 1, 0, -4035626, 3117787680]\) | \(154502321244119857/55101928644\) | \(2592318777856115364\) | \([2, 2]\) | \(13824000\) | \(2.5026\) | |
348726.b4 | 348726b1 | \([1, 1, 0, -216246, 63047556]\) | \(-23771111713777/22848457968\) | \(-1074925834596029808\) | \([2]\) | \(6912000\) | \(2.1561\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 348726.b have rank \(1\).
Complex multiplication
The elliptic curves in class 348726.b do not have complex multiplication.Modular form 348726.2.a.b
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.