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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 4186.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4186.a1 | 4186a2 | \([1, 1, 1, -7826, -269729]\) | \(53008645999484449/2060047808\) | \(2060047808\) | \([2]\) | \(9984\) | \(0.87170\) | |
4186.a2 | 4186a1 | \([1, 1, 1, -466, -4769]\) | \(-11192824869409/2563305472\) | \(-2563305472\) | \([2]\) | \(4992\) | \(0.52513\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4186.a have rank \(0\).
Complex multiplication
The elliptic curves in class 4186.a do not have complex multiplication.Modular form 4186.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.