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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 51870.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51870.p1 | 51870p1 | \([1, 1, 0, -48167, 4048821]\) | \(12359092816971484921/116188800000\) | \(116188800000\) | \([2]\) | \(256000\) | \(1.2861\) | \(\Gamma_0(N)\)-optimal |
51870.p2 | 51870p2 | \([1, 1, 0, -47047, 4247509]\) | \(-11516856136356002041/1201114687500000\) | \(-1201114687500000\) | \([2]\) | \(512000\) | \(1.6326\) |
Rank
sage: E.rank()
The elliptic curves in class 51870.p have rank \(1\).
Complex multiplication
The elliptic curves in class 51870.p do not have complex multiplication.Modular form 51870.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.