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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 1290.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1290.c1 | 1290e4 | \([1, 0, 1, -66374, 6575672]\) | \(32337636827233520089/3023437500000\) | \(3023437500000\) | \([2]\) | \(7680\) | \(1.4326\) | |
1290.c2 | 1290e3 | \([1, 0, 1, -24454, -1401544]\) | \(1617141066657115609/89723013444000\) | \(89723013444000\) | \([2]\) | \(7680\) | \(1.4326\) | |
1290.c3 | 1290e2 | \([1, 0, 1, -4454, 86456]\) | \(9768641617435609/2396304000000\) | \(2396304000000\) | \([2, 2]\) | \(3840\) | \(1.0860\) | |
1290.c4 | 1290e1 | \([1, 0, 1, 666, 8632]\) | \(32740359775271/50724864000\) | \(-50724864000\) | \([2]\) | \(1920\) | \(0.73942\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1290.c have rank \(1\).
Complex multiplication
The elliptic curves in class 1290.c do not have complex multiplication.Modular form 1290.2.a.c
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.