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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 7623.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7623.k1 | 7623e2 | \([0, 0, 1, -3036, -64386]\) | \(35084566528/1029\) | \(90767061\) | \([]\) | \(4608\) | \(0.62679\) | |
7623.k2 | 7623e1 | \([0, 0, 1, -66, 63]\) | \(360448/189\) | \(16671501\) | \([]\) | \(1536\) | \(0.077485\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 7623.k have rank \(1\).
Complex multiplication
The elliptic curves in class 7623.k do not have complex multiplication.Modular form 7623.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.