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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 157794x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
157794.bq3 | 157794x1 | \([1, 1, 1, -2029, -32365]\) | \(38272753/4368\) | \(105432901392\) | \([2]\) | \(221184\) | \(0.84743\) | \(\Gamma_0(N)\)-optimal |
157794.bq2 | 157794x2 | \([1, 1, 1, -7809, 228891]\) | \(2181825073/298116\) | \(7195795520004\) | \([2, 2]\) | \(442368\) | \(1.1940\) | |
157794.bq1 | 157794x3 | \([1, 1, 1, -120519, 16053375]\) | \(8020417344913/187278\) | \(4520435647182\) | \([2]\) | \(884736\) | \(1.5406\) | |
157794.bq4 | 157794x4 | \([1, 1, 1, 12421, 1240391]\) | \(8780064047/32388174\) | \(-781771784709006\) | \([2]\) | \(884736\) | \(1.5406\) |
Rank
sage: E.rank()
The elliptic curves in class 157794x have rank \(0\).
Complex multiplication
The elliptic curves in class 157794x do not have complex multiplication.Modular form 157794.2.a.x
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.