Properties

Label 258570cw
Number of curves $4$
Conductor $258570$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("cw1")
 
E.isogeny_class()
 

Elliptic curves in class 258570cw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
258570.cw3 258570cw1 \([1, -1, 0, -893619, -324472667]\) \(22428153804601/35802000\) \(125978064131322000\) \([2]\) \(7225344\) \(2.1789\) \(\Gamma_0(N)\)-optimal
258570.cw2 258570cw2 \([1, -1, 0, -1167399, -108898295]\) \(50002789171321/27473062500\) \(96670667267438062500\) \([2, 2]\) \(14450688\) \(2.5255\)  
258570.cw1 258570cw3 \([1, -1, 0, -11251629, 14438611903]\) \(44769506062996441/323730468750\) \(1139124567159667968750\) \([2]\) \(28901376\) \(2.8720\)  
258570.cw4 258570cw4 \([1, -1, 0, 4536351, -862934045]\) \(2933972022568679/1789082460750\) \(-6295322746678589880750\) \([2]\) \(28901376\) \(2.8720\)  

Rank

sage: E.rank()
 

The elliptic curves in class 258570cw have rank \(0\).

Complex multiplication

The elliptic curves in class 258570cw do not have complex multiplication.

Modular form 258570.2.a.cw

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