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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 122694be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122694.bh4 | 122694be1 | \([1, 0, 1, 7350989, -5515298098]\) | \(5137417856375/4510142208\) | \(-38566165489308394958592\) | \([2]\) | \(11612160\) | \(3.0214\) | \(\Gamma_0(N)\)-optimal |
122694.bh3 | 122694be2 | \([1, 0, 1, -36818851, -48996088594]\) | \(645532578015625/252306960048\) | \(2157473429120638844824752\) | \([2]\) | \(23224320\) | \(3.3679\) | |
122694.bh2 | 122694be3 | \([1, 0, 1, -76387666, 343138965860]\) | \(-5764706497797625/2612665516032\) | \(-22340867762611228841607168\) | \([2]\) | \(34836480\) | \(3.5707\) | |
122694.bh1 | 122694be4 | \([1, 0, 1, -1332774226, 18725582002532]\) | \(30618029936661765625/3678951124992\) | \(31458661694048021862494208\) | \([2]\) | \(69672960\) | \(3.9172\) |
Rank
sage: E.rank()
The elliptic curves in class 122694be have rank \(1\).
Complex multiplication
The elliptic curves in class 122694be do not have complex multiplication.Modular form 122694.2.a.be
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.