Properties

Label 400710i
Number of curves $2$
Conductor $400710$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 400710i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
400710.i2 400710i1 \([1, 1, 0, -212997, 38176461]\) \(-22715680520161/299646720\) \(-14097143931160320\) \([2]\) \(4055040\) \(1.9069\) \(\Gamma_0(N)\)-optimal*
400710.i1 400710i2 \([1, 1, 0, -3418677, 2431537149]\) \(93923986054560481/16028400\) \(754070199020400\) \([2]\) \(8110080\) \(2.2535\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 400710i1.

Rank

sage: E.rank()
 

The elliptic curves in class 400710i have rank \(1\).

Complex multiplication

The elliptic curves in class 400710i do not have complex multiplication.

Modular form 400710.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2 q^{7} - q^{8} + q^{9} - q^{10} - 4 q^{11} - q^{12} + 2 q^{13} + 2 q^{14} - q^{15} + q^{16} - 2 q^{17} - q^{18} + O(q^{20})\) 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.