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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 5010e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5010.e3 | 5010e1 | \([1, 1, 1, -471, -4131]\) | \(11556972012529/360720\) | \(360720\) | \([2]\) | \(1824\) | \(0.16274\) | \(\Gamma_0(N)\)-optimal |
5010.e2 | 5010e2 | \([1, 1, 1, -491, -3787]\) | \(13092526729009/2033108100\) | \(2033108100\) | \([2, 2]\) | \(3648\) | \(0.50931\) | |
5010.e1 | 5010e3 | \([1, 1, 1, -2161, 34289]\) | \(1116093485689489/110938308750\) | \(110938308750\) | \([2]\) | \(7296\) | \(0.85589\) | |
5010.e4 | 5010e4 | \([1, 1, 1, 859, -19447]\) | \(70092508729391/210005006670\) | \(-210005006670\) | \([2]\) | \(7296\) | \(0.85589\) |
Rank
sage: E.rank()
The elliptic curves in class 5010e have rank \(0\).
Complex multiplication
The elliptic curves in class 5010e do not have complex multiplication.Modular form 5010.2.a.e
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.