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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 74529.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
74529.p1 | 74529s2 | \([0, 0, 1, -223860, -40767386]\) | \(-205514702848000/27\) | \(-162994923\) | \([]\) | \(191808\) | \(1.4336\) | |
74529.p2 | 74529s1 | \([0, 0, 1, -2730, -57353]\) | \(-372736000/19683\) | \(-118823298867\) | \([]\) | \(63936\) | \(0.88434\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 74529.p have rank \(1\).
Complex multiplication
The elliptic curves in class 74529.p do not have complex multiplication.Modular form 74529.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.