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SageMath
E = EllipticCurve("pz1")
E.isogeny_class()
Elliptic curves in class 141120pz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
141120.hi1 | 141120pz1 | \([0, 0, 0, -47628, -1926288]\) | \(78732/35\) | \(5311626238033920\) | \([2]\) | \(884736\) | \(1.7130\) | \(\Gamma_0(N)\)-optimal |
141120.hi2 | 141120pz2 | \([0, 0, 0, 164052, -14373072]\) | \(1608714/1225\) | \(-371813836662374400\) | \([2]\) | \(1769472\) | \(2.0596\) |
Rank
sage: E.rank()
The elliptic curves in class 141120pz have rank \(0\).
Complex multiplication
The elliptic curves in class 141120pz do not have complex multiplication.Modular form 141120.2.a.pz
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.