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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 305045.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
305045.k1 | 305045k2 | \([0, -1, 1, -49235, 3147523]\) | \(7575076864/1953125\) | \(3403277439453125\) | \([]\) | \(1741824\) | \(1.6896\) | |
305045.k2 | 305045k1 | \([0, -1, 1, -17125, -856594]\) | \(318767104/125\) | \(217809756125\) | \([]\) | \(580608\) | \(1.1402\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 305045.k have rank \(1\).
Complex multiplication
The elliptic curves in class 305045.k do not have complex multiplication.Modular form 305045.2.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.