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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 126075.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
126075.a1 | 126075l2 | \([0, -1, 1, -350208, -81647182]\) | \(-102400/3\) | \(-139163210185546875\) | \([]\) | \(2040000\) | \(2.0684\) | |
126075.a2 | 126075l1 | \([0, -1, 1, 2802, 251138]\) | \(20480/243\) | \(-28856883264075\) | \([]\) | \(408000\) | \(1.2637\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 126075.a have rank \(1\).
Complex multiplication
The elliptic curves in class 126075.a do not have complex multiplication.Modular form 126075.2.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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.