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Properties

Label 436800ho
Number of curves $4$
Conductor $436800$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ho1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 436800ho have rank \(1\).

Complex multiplication

The elliptic curves in class 436800ho do not have complex multiplication.

Modular form 436800.2.a.ho

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{7} + q^{9} - q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 436800ho

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
436800.ho3 436800ho1 \([0, -1, 0, -509497633, 3516950867137]\) \(3571003510905229697089/762141946675200000\) \(3121733413581619200000000000\) \([2]\) \(212336640\) \(3.9900\) \(\Gamma_0(N)\)-optimal
436800.ho2 436800ho2 \([0, -1, 0, -2606649633, -48146388652863]\) \(478202393398338853167169/32244226560000000000\) \(132072351989760000000000000000\) \([2, 2]\) \(424673280\) \(4.3366\)  
436800.ho4 436800ho3 \([0, -1, 0, 2238918367, -206640072364863]\) \(303025056761573589385151/4678857421875000000000\) \(-19164600000000000000000000000000\) \([2]\) \(849346560\) \(4.6832\)  
436800.ho1 436800ho4 \([0, -1, 0, -41006649633, -3196139988652863]\) \(1861772567578966373029167169/9401133413380800000\) \(38507042461207756800000000000\) \([2]\) \(849346560\) \(4.6832\)