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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 75810.cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75810.cr1 | 75810cy4 | \([1, 0, 0, -134841, -19069389]\) | \(5763259856089/5670\) | \(266750145270\) | \([2]\) | \(460800\) | \(1.4858\) | |
75810.cr2 | 75810cy2 | \([1, 0, 0, -8491, -293779]\) | \(1439069689/44100\) | \(2074723352100\) | \([2, 2]\) | \(230400\) | \(1.1392\) | |
75810.cr3 | 75810cy1 | \([1, 0, 0, -1271, 10905]\) | \(4826809/1680\) | \(79037080080\) | \([2]\) | \(115200\) | \(0.79265\) | \(\Gamma_0(N)\)-optimal |
75810.cr4 | 75810cy3 | \([1, 0, 0, 2339, -989065]\) | \(30080231/9003750\) | \(-423589351053750\) | \([2]\) | \(460800\) | \(1.4858\) |
Rank
sage: E.rank()
The elliptic curves in class 75810.cr have rank \(0\).
Complex multiplication
The elliptic curves in class 75810.cr do not have complex multiplication.Modular form 75810.2.a.cr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.