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SageMath
E = EllipticCurve("hl1")
E.isogeny_class()
Elliptic curves in class 433200.hl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
433200.hl1 | 433200hl1 | \([0, 1, 0, -5201408, -5259220812]\) | \(-14317849/2700\) | \(-2934759693484800000000\) | \([]\) | \(28366848\) | \(2.8434\) | \(\Gamma_0(N)\)-optimal* |
433200.hl2 | 433200hl2 | \([0, 1, 0, 35952592, 25770895188]\) | \(4728305591/3000000\) | \(-3260844103872000000000000\) | \([]\) | \(85100544\) | \(3.3927\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 433200.hl have rank \(1\).
Complex multiplication
The elliptic curves in class 433200.hl do not have complex multiplication.Modular form 433200.2.a.hl
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.