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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 450.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450.a1 | 450f4 | \([1, -1, 0, -5442, -39034]\) | \(57960603/31250\) | \(9610839843750\) | \([2]\) | \(1152\) | \(1.1822\) | |
450.a2 | 450f2 | \([1, -1, 0, -3192, 70216]\) | \(8527173507/200\) | \(84375000\) | \([2]\) | \(384\) | \(0.63293\) | |
450.a3 | 450f1 | \([1, -1, 0, -192, 1216]\) | \(-1860867/320\) | \(-135000000\) | \([2]\) | \(192\) | \(0.28636\) | \(\Gamma_0(N)\)-optimal |
450.a4 | 450f3 | \([1, -1, 0, 1308, -5284]\) | \(804357/500\) | \(-153773437500\) | \([2]\) | \(576\) | \(0.83566\) |
Rank
sage: E.rank()
The elliptic curves in class 450.a have rank \(1\).
Complex multiplication
The elliptic curves in class 450.a do not have complex multiplication.Modular form 450.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.