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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 69828ba
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69828.bd2 | 69828ba1 | \([0, 1, 0, -87461, 4118892]\) | \(31238127616/14939397\) | \(35385070656302928\) | \([2]\) | \(760320\) | \(1.8693\) | \(\Gamma_0(N)\)-optimal |
69828.bd1 | 69828ba2 | \([0, 1, 0, -730196, -237549468]\) | \(1136150003536/15554187\) | \(589459942455614208\) | \([2]\) | \(1520640\) | \(2.2159\) |
Rank
sage: E.rank()
The elliptic curves in class 69828ba have rank \(0\).
Complex multiplication
The elliptic curves in class 69828ba do not have complex multiplication.Modular form 69828.2.a.ba
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.