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Properties

Label 333270.c
Number of curves $2$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 333270.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.c1 333270c2 \([1, -1, 0, -58089060, 170299441066]\) \(16509301927847/13781250\) \(18095336499408700218750\) \([2]\) \(67829760\) \(3.1988\)  
333270.c2 333270c1 \([1, -1, 0, -4432590, 1399604800]\) \(7335308807/3601500\) \(4728914605178806990500\) \([2]\) \(33914880\) \(2.8522\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 333270.c have rank \(1\).

Complex multiplication

The elliptic curves in class 333270.c do not have complex multiplication.

Modular form 333270.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + q^{10} - 4 q^{11} - 4 q^{13} + q^{14} + q^{16} + 8 q^{17} - 6 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.