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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 12675y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12675.m2 | 12675y1 | \([1, 0, 0, -42338, 3388917]\) | \(-658489/9\) | \(-114712132640625\) | \([]\) | \(43680\) | \(1.5046\) | \(\Gamma_0(N)\)-optimal |
12675.m1 | 12675y2 | \([1, 0, 0, -316963, -381909958]\) | \(-276301129/4782969\) | \(-60962730482666390625\) | \([]\) | \(305760\) | \(2.4776\) |
Rank
sage: E.rank()
The elliptic curves in class 12675y have rank \(0\).
Complex multiplication
The elliptic curves in class 12675y do not have complex multiplication.Modular form 12675.2.a.y
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.