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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1805.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1805.a1 | 1805a2 | \([0, 1, 1, -105171, -9810339]\) | \(7575076864/1953125\) | \(33171021564453125\) | \([]\) | \(14364\) | \(1.8793\) | |
1805.a2 | 1805a1 | \([0, 1, 1, -36581, 2679900]\) | \(318767104/125\) | \(2122945380125\) | \([3]\) | \(4788\) | \(1.3300\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1805.a have rank \(1\).
Complex multiplication
The elliptic curves in class 1805.a do not have complex multiplication.Modular form 1805.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.