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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 73034.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
73034.k1 | 73034k3 | \([1, 1, 1, -1290794, -564998601]\) | \(-10730978619193/6656\) | \(-147525987674624\) | \([]\) | \(909792\) | \(2.0395\) | |
73034.k2 | 73034k2 | \([1, 1, 1, -12699, -1103087]\) | \(-10218313/17576\) | \(-389560811203304\) | \([]\) | \(303264\) | \(1.4902\) | |
73034.k3 | 73034k1 | \([1, 1, 1, 1346, 31749]\) | \(12167/26\) | \(-576273389354\) | \([]\) | \(101088\) | \(0.94092\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 73034.k have rank \(1\).
Complex multiplication
The elliptic curves in class 73034.k do not have complex multiplication.Modular form 73034.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.