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Properties

Label 250470p
Number of curves $2$
Conductor $250470$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 250470p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
250470.p1 250470p1 \([1, -1, 0, -4636440, 3801960256]\) \(8534813931497881/106867200000\) \(138015565736716800000\) \([2]\) \(9216000\) \(2.6740\) \(\Gamma_0(N)\)-optimal
250470.p2 250470p2 \([1, -1, 0, -803160, 9896108800]\) \(-44365623586201/32731875000000\) \(-42272168127811875000000\) \([2]\) \(18432000\) \(3.0206\)  

Rank

sage: E.rank()
 

The elliptic curves in class 250470p have rank \(1\).

Complex multiplication

The elliptic curves in class 250470p do not have complex multiplication.

Modular form 250470.2.a.p

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.