Properties

Label 304920.cz
Number of curves $4$
Conductor $304920$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cz1")
 
E.isogeny_class()
 

Elliptic curves in class 304920.cz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
304920.cz1 304920cz4 \([0, 0, 0, -17744924307, -805818360558706]\) \(233632133015204766393938/29145526885986328125\) \(77087773517577539062500000000000\) \([2]\) \(943718400\) \(4.8482\)  
304920.cz2 304920cz2 \([0, 0, 0, -4432422027, 100542057171446]\) \(7282213870869695463556/912102595400390625\) \(1206222117275004131610000000000\) \([2, 2]\) \(471859200\) \(4.5016\)  
304920.cz3 304920cz1 \([0, 0, 0, -4289523447, 108132058088714]\) \(26401417552259125806544/507547744790625\) \(167803303714631372541600000\) \([2]\) \(235929600\) \(4.1550\) \(\Gamma_0(N)\)-optimal
304920.cz4 304920cz3 \([0, 0, 0, 6593702973, 521142416196446]\) \(11986661998777424518222/51295853620928503125\) \(-135673757478792867481710393600000\) \([2]\) \(943718400\) \(4.8482\)  

Rank

sage: E.rank()
 

The elliptic curves in class 304920.cz have rank \(0\).

Complex multiplication

The elliptic curves in class 304920.cz do not have complex multiplication.

Modular form 304920.2.a.cz

sage: E.q_eigenform(10)
 
\(q + q^{5} - q^{7} - 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.