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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 138.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
138.c1 | 138c4 | \([1, 1, 1, -247, 1391]\) | \(1666957239793/301806\) | \(301806\) | \([2]\) | \(32\) | \(0.055014\) | |
138.c2 | 138c3 | \([1, 1, 1, -107, -457]\) | \(135559106353/5037138\) | \(5037138\) | \([2]\) | \(32\) | \(0.055014\) | |
138.c3 | 138c2 | \([1, 1, 1, -17, 11]\) | \(545338513/171396\) | \(171396\) | \([2, 2]\) | \(16\) | \(-0.29156\) | |
138.c4 | 138c1 | \([1, 1, 1, 3, 3]\) | \(2924207/3312\) | \(-3312\) | \([4]\) | \(8\) | \(-0.63813\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 138.c have rank \(0\).
Complex multiplication
The elliptic curves in class 138.c do not have complex multiplication.Modular form 138.2.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.