Properties

Label 466578.et
Number of curves $2$
Conductor $466578$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("et1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 466578.et

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
466578.et1 466578et2 \([1, -1, 1, -141261350, -604961710027]\) \(24553362849625/1755162752\) \(22284360639299350053237888\) \([2]\) \(136249344\) \(3.6108\) \(\Gamma_0(N)\)-optimal*
466578.et2 466578et1 \([1, -1, 1, 8043610, -41305625035]\) \(4533086375/60669952\) \(-770293859527495762034688\) \([2]\) \(68124672\) \(3.2642\) \(\Gamma_0(N)\)-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 466578.et1.

Rank

sage: E.rank()
 

The elliptic curves in class 466578.et have rank \(1\).

Complex multiplication

The elliptic curves in class 466578.et do not have complex multiplication.

Modular form 466578.2.a.et

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{8} + 4q^{11} + q^{16} - 6q^{17} - 6q^{19} + O(q^{20})\)  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 LMFDB 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 LMFDB labels.