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SageMath
E = EllipticCurve("ok1")
E.isogeny_class()
Elliptic curves in class 446400ok
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
446400.ok4 | 446400ok1 | \([0, 0, 0, 20001300, -38629006000]\) | \(296354077829711/387386634240\) | \(-1156730291654492160000000\) | \([2]\) | \(53084160\) | \(3.3032\) | \(\Gamma_0(N)\)-optimal* |
446400.ok3 | 446400ok2 | \([0, 0, 0, -122846700, -376607374000]\) | \(68663623745397169/19216056254400\) | \(57378836518738329600000000\) | \([2]\) | \(106168320\) | \(3.6498\) | \(\Gamma_0(N)\)-optimal* |
446400.ok2 | 446400ok3 | \([0, 0, 0, -570974700, -5282825614000]\) | \(-6894246873502147249/47925198774000\) | \(-143103876735983616000000000\) | \([2]\) | \(159252480\) | \(3.8525\) | |
446400.ok1 | 446400ok4 | \([0, 0, 0, -9150782700, -336926724046000]\) | \(28379906689597370652529/1357352437500\) | \(4053032660736000000000000\) | \([2]\) | \(318504960\) | \(4.1991\) |
Rank
sage: E.rank()
The elliptic curves in class 446400ok have rank \(1\).
Complex multiplication
The elliptic curves in class 446400ok do not have complex multiplication.Modular form 446400.2.a.ok
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.