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Properties

Label 286650.ha
Number of curves $2$
Conductor $286650$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ha1")
 
E.isogeny_class()
 

Elliptic curves in class 286650.ha

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
286650.ha1 286650ha2 \([1, -1, 0, -873115917, 4971186752741]\) \(2034416504287874043/882294347833600\) \(31923687850556888868300000000\) \([2]\) \(265420800\) \(4.1656\)  
286650.ha2 286650ha1 \([1, -1, 0, 185284083, 575651552741]\) \(19441890357117957/15208161280000\) \(-550270547097747840000000000\) \([2]\) \(132710400\) \(3.8190\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 286650.ha have rank \(0\).

Complex multiplication

The elliptic curves in class 286650.ha do not have complex multiplication.

Modular form 286650.2.a.ha

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{8} + 4 q^{11} - q^{13} + q^{16} - 4 q^{17} - 6 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.