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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 74970.dk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
74970.dk1 | 74970df2 | \([1, -1, 1, -23792, 864861]\) | \(5956317014383/2172381210\) | \(543197404416870\) | \([2]\) | \(294912\) | \(1.5278\) | |
74970.dk2 | 74970df1 | \([1, -1, 1, 4558, 93741]\) | \(41890384817/39795300\) | \(-9950695379100\) | \([2]\) | \(147456\) | \(1.1812\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 74970.dk have rank \(1\).
Complex multiplication
The elliptic curves in class 74970.dk do not have complex multiplication.Modular form 74970.2.a.dk
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.