Properties

Label 433200.hl
Number of curves $2$
Conductor $433200$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("hl1")
 
E.isogeny_class()
 

Elliptic curves in class 433200.hl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
433200.hl1 433200hl1 \([0, 1, 0, -5201408, -5259220812]\) \(-14317849/2700\) \(-2934759693484800000000\) \([]\) \(28366848\) \(2.8434\) \(\Gamma_0(N)\)-optimal*
433200.hl2 433200hl2 \([0, 1, 0, 35952592, 25770895188]\) \(4728305591/3000000\) \(-3260844103872000000000000\) \([]\) \(85100544\) \(3.3927\) \(\Gamma_0(N)\)-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 433200.hl1.

Rank

sage: E.rank()
 

The elliptic curves in class 433200.hl have rank \(1\).

Complex multiplication

The elliptic curves in class 433200.hl do not have complex multiplication.

Modular form 433200.2.a.hl

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{7} + q^{9} - 6 q^{11} - 5 q^{13} + O(q^{20})\) Copy content 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.