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Properties

Label 470400.ke
Number of curves $2$
Conductor $470400$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ke1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 470400.ke have rank \(1\).

Complex multiplication

The elliptic curves in class 470400.ke do not have complex multiplication.

Modular form 470400.2.a.ke

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + q^{9} - 2 q^{13} - 4 q^{17} - 2 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.

Elliptic curves in class 470400.ke

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
470400.ke1 470400ke2 \([0, 1, 0, -815033, 282843063]\) \(3976047968/1575\) \(23718038400000000\) \([2]\) \(4718592\) \(2.1063\) \(\Gamma_0(N)\)-optimal*
470400.ke2 470400ke1 \([0, 1, 0, -43283, 5784813]\) \(-19056256/19845\) \(-9338977620000000\) \([2]\) \(2359296\) \(1.7597\) \(\Gamma_0(N)\)-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 470400.ke1.