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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 31939.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31939.e1 | 31939b3 | \([0, -1, 1, -1293249, -565640702]\) | \(-50357871050752/19\) | \(-90251980579\) | \([]\) | \(207360\) | \(1.8902\) | |
31939.e2 | 31939b2 | \([0, -1, 1, -15689, -799487]\) | \(-89915392/6859\) | \(-32580964989019\) | \([]\) | \(69120\) | \(1.3409\) | |
31939.e3 | 31939b1 | \([0, -1, 1, 1121, -1012]\) | \(32768/19\) | \(-90251980579\) | \([]\) | \(23040\) | \(0.79161\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 31939.e have rank \(1\).
Complex multiplication
The elliptic curves in class 31939.e do not have complex multiplication.Modular form 31939.2.a.e
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.