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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 463680.cb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.cb1 | 463680cb4 | \([0, 0, 0, -249281868, -1514900635792]\) | \(8964546681033941529169/31696875000\) | \(6057367142400000000\) | \([2]\) | \(56623104\) | \(3.2463\) | |
463680.cb2 | 463680cb3 | \([0, 0, 0, -20771148, -6563498128]\) | \(5186062692284555089/2903809817953800\) | \(554926697948989410508800\) | \([2]\) | \(56623104\) | \(3.2463\) | \(\Gamma_0(N)\)-optimal* |
463680.cb3 | 463680cb2 | \([0, 0, 0, -15587148, -23647888528]\) | \(2191574502231419089/4115217960000\) | \(786430399044648960000\) | \([2, 2]\) | \(28311552\) | \(2.8997\) | \(\Gamma_0(N)\)-optimal* |
463680.cb4 | 463680cb1 | \([0, 0, 0, -657228, -614007952]\) | \(-164287467238609/757170892800\) | \(-144697610954656972800\) | \([2]\) | \(14155776\) | \(2.5532\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 463680.cb have rank \(1\).
Complex multiplication
The elliptic curves in class 463680.cb do not have complex multiplication.Modular form 463680.2.a.cb
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.