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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 85800a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
85800.t2 | 85800a1 | \([0, -1, 0, 148192, 37143612]\) | \(22494434350748/50367250791\) | \(-805876012656000000\) | \([2]\) | \(1204224\) | \(2.1200\) | \(\Gamma_0(N)\)-optimal |
85800.t1 | 85800a2 | \([0, -1, 0, -1182808, 407161612]\) | \(5718957389087906/1075876263891\) | \(34428040444512000000\) | \([2]\) | \(2408448\) | \(2.4666\) |
Rank
sage: E.rank()
The elliptic curves in class 85800a have rank \(1\).
Complex multiplication
The elliptic curves in class 85800a do not have complex multiplication.Modular form 85800.2.a.a
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.