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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 198198.ec
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
198198.ec1 | 198198ba2 | \([1, -1, 1, -4001526167, -97427631773887]\) | \(-5486773802537974663600129/2635437714\) | \(-3403583391925582866\) | \([]\) | \(78368640\) | \(3.7948\) | |
198198.ec2 | 198198ba1 | \([1, -1, 1, 777523, -2981797387]\) | \(40251338884511/2997011332224\) | \(-3870544138297313533056\) | \([]\) | \(11195520\) | \(2.8218\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 198198.ec have rank \(1\).
Complex multiplication
The elliptic curves in class 198198.ec do not have complex multiplication.Modular form 198198.2.a.ec
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.