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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 51376x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51376.w3 | 51376x1 | \([0, -1, 0, 1803, 701]\) | \(32768/19\) | \(-375641583616\) | \([]\) | \(51840\) | \(0.91045\) | \(\Gamma_0(N)\)-optimal |
51376.w2 | 51376x2 | \([0, -1, 0, -25237, 1650141]\) | \(-89915392/6859\) | \(-135606611685376\) | \([]\) | \(155520\) | \(1.4598\) | |
51376.w1 | 51376x3 | \([0, -1, 0, -2080277, 1155555101]\) | \(-50357871050752/19\) | \(-375641583616\) | \([]\) | \(466560\) | \(2.0091\) |
Rank
sage: E.rank()
The elliptic curves in class 51376x have rank \(1\).
Complex multiplication
The elliptic curves in class 51376x do not have complex multiplication.Modular form 51376.2.a.x
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.