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Properties

Label 458640.hn
Number of curves $4$
Conductor $458640$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("hn1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 458640.hn have rank \(0\).

Complex multiplication

The elliptic curves in class 458640.hn do not have complex multiplication.

Modular form 458640.2.a.hn

Copy content sage:E.q_eigenform(10)
 
\(q + q^{5} - 6 q^{11} - q^{13} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 458640.hn

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.hn1 458640hn4 \([0, 0, 0, -112837347, 423391378914]\) \(16751080718799363/1529437000000\) \(14506781604678365184000000\) \([2]\) \(95551488\) \(3.5672\)  
458640.hn2 458640hn2 \([0, 0, 0, -110238387, 445499671666]\) \(11387025941627437947/10765300\) \(140067729620582400\) \([2]\) \(31850496\) \(3.0179\) \(\Gamma_0(N)\)-optimal*
458640.hn3 458640hn3 \([0, 0, 0, -24778467, -40027282974]\) \(177381177331203/29679104000\) \(281507692013816905728000\) \([2]\) \(47775744\) \(3.2206\)  
458640.hn4 458640hn1 \([0, 0, 0, -6891507, 6957521074]\) \(2781982314427707/2703013040\) \(35169005940162232320\) \([2]\) \(15925248\) \(2.6713\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 458640.hn1.