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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 80850.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
80850.b1 | 80850o4 | \([1, 1, 0, -431225, 108814875]\) | \(4824238966273/66\) | \(121325531250\) | \([2]\) | \(589824\) | \(1.6827\) | |
80850.b2 | 80850o2 | \([1, 1, 0, -26975, 1688625]\) | \(1180932193/4356\) | \(8007485062500\) | \([2, 2]\) | \(294912\) | \(1.3361\) | |
80850.b3 | 80850o3 | \([1, 1, 0, -14725, 3244375]\) | \(-192100033/2371842\) | \(-4360075616531250\) | \([2]\) | \(589824\) | \(1.6827\) | |
80850.b4 | 80850o1 | \([1, 1, 0, -2475, -1875]\) | \(912673/528\) | \(970604250000\) | \([2]\) | \(147456\) | \(0.98957\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 80850.b have rank \(2\).
Complex multiplication
The elliptic curves in class 80850.b do not have complex multiplication.Modular form 80850.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.