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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 3696x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3696.p2 | 3696x1 | \([0, 1, 0, -6480, -315756]\) | \(-7347774183121/6119866368\) | \(-25066972643328\) | \([2]\) | \(16128\) | \(1.2689\) | \(\Gamma_0(N)\)-optimal |
3696.p1 | 3696x2 | \([0, 1, 0, -119120, -15860076]\) | \(45637459887836881/13417633152\) | \(54958625390592\) | \([2]\) | \(32256\) | \(1.6155\) |
Rank
sage: E.rank()
The elliptic curves in class 3696x have rank \(0\).
Complex multiplication
The elliptic curves in class 3696x do not have complex multiplication.Modular form 3696.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.