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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 220.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
220.a1 | 220a4 | \([0, 1, 0, -7100, -232652]\) | \(154639330142416/33275\) | \(8518400\) | \([2]\) | \(216\) | \(0.71445\) | |
220.a2 | 220a3 | \([0, 1, 0, -445, -3720]\) | \(610462990336/8857805\) | \(141724880\) | \([2]\) | \(108\) | \(0.36788\) | |
220.a3 | 220a2 | \([0, 1, 0, -100, -252]\) | \(436334416/171875\) | \(44000000\) | \([6]\) | \(72\) | \(0.16514\) | |
220.a4 | 220a1 | \([0, 1, 0, -45, 100]\) | \(643956736/15125\) | \(242000\) | \([6]\) | \(36\) | \(-0.18143\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 220.a have rank \(1\).
Complex multiplication
The elliptic curves in class 220.a do not have complex multiplication.Modular form 220.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.