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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 85176ce
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
85176.bs4 | 85176ce1 | \([0, 0, 0, 1521, -118638]\) | \(432/7\) | \(-6305588819712\) | \([2]\) | \(147456\) | \(1.1366\) | \(\Gamma_0(N)\)-optimal |
85176.bs3 | 85176ce2 | \([0, 0, 0, -28899, -1779570]\) | \(740772/49\) | \(176556486951936\) | \([2, 2]\) | \(294912\) | \(1.4832\) | |
85176.bs2 | 85176ce3 | \([0, 0, 0, -89739, 8186022]\) | \(11090466/2401\) | \(17302535721289728\) | \([2]\) | \(589824\) | \(1.8297\) | |
85176.bs1 | 85176ce4 | \([0, 0, 0, -454779, -118044810]\) | \(1443468546/7\) | \(50444710557696\) | \([2]\) | \(589824\) | \(1.8297\) |
Rank
sage: E.rank()
The elliptic curves in class 85176ce have rank \(0\).
Complex multiplication
The elliptic curves in class 85176ce do not have complex multiplication.Modular form 85176.2.a.ce
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.