↑

Properties

Label 400078e
Number of curves $2$
Conductor $400078$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("e1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 400078e have rank \(1\).

Complex multiplication

The elliptic curves in class 400078e do not have complex multiplication.

Modular form 400078.2.a.e

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{5} + q^{7} - q^{8} - 3 q^{9} + 2 q^{10} + 2 q^{11} - q^{14} + q^{16} + q^{17} + 3 q^{18} + 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 400078e

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
400078.e2 400078e1 \([1, -1, 0, 3047, 31545]\) \(658503/476\) \(-2261049618716\) \([2]\) \(552960\) \(1.0589\) \(\Gamma_0(N)\)-optimal*
400078.e1 400078e2 \([1, -1, 0, -13763, 276971]\) \(60698457/28322\) \(134532452313602\) \([2]\) \(1105920\) \(1.4055\) \(\Gamma_0(N)\)-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 400078e1.