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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 414.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414.a1 | 414c3 | \([1, -1, 0, -2223, -39785]\) | \(1666957239793/301806\) | \(220016574\) | \([2]\) | \(256\) | \(0.60432\) | |
414.a2 | 414c4 | \([1, -1, 0, -963, 11371]\) | \(135559106353/5037138\) | \(3672073602\) | \([2]\) | \(256\) | \(0.60432\) | |
414.a3 | 414c2 | \([1, -1, 0, -153, -455]\) | \(545338513/171396\) | \(124947684\) | \([2, 2]\) | \(128\) | \(0.25775\) | |
414.a4 | 414c1 | \([1, -1, 0, 27, -59]\) | \(2924207/3312\) | \(-2414448\) | \([2]\) | \(64\) | \(-0.088827\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 414.a have rank \(1\).
Complex multiplication
The elliptic curves in class 414.a do not have complex multiplication.Modular form 414.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.