Show commands:
SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 471510.cz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
471510.cz1 | 471510cz4 | \([1, -1, 1, -966551423, 11566304336747]\) | \(28379906689597370652529/1357352437500\) | \(4776175420931267437500\) | \([2]\) | \(149299200\) | \(3.6371\) | \(\Gamma_0(N)\)-optimal* |
471510.cz2 | 471510cz3 | \([1, -1, 1, -60309203, 181364575331]\) | \(-6894246873502147249/47925198774000\) | \(-168636494180697349014000\) | \([2]\) | \(74649600\) | \(3.2905\) | \(\Gamma_0(N)\)-optimal* |
471510.cz3 | 471510cz2 | \([1, -1, 1, -12975683, 12931468931]\) | \(68663623745397169/19216056254400\) | \(67616378056195028798400\) | \([2]\) | \(49766400\) | \(3.0878\) | \(\Gamma_0(N)\)-optimal* |
471510.cz4 | 471510cz1 | \([1, -1, 1, 2112637, 1325533187]\) | \(296354077829711/387386634240\) | \(-1363114302326788976640\) | \([2]\) | \(24883200\) | \(2.7412\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 471510.cz have rank \(1\).
Complex multiplication
The elliptic curves in class 471510.cz do not have complex multiplication.Modular form 471510.2.a.cz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.