Properties

Label 286650.ho
Number of curves $4$
Conductor $286650$
CM no
Rank $0$
Graph

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

Elliptic curves in class 286650.ho

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
286650.ho1 286650ho3 \([1, -1, 0, -86474817, 309537043591]\) \(53365044437418169/41984670\) \(56263473240079218750\) \([2]\) \(28311552\) \(3.0963\)  
286650.ho2 286650ho4 \([1, -1, 0, -12607317, -10416173909]\) \(165369706597369/60703354530\) \(81348300776966845781250\) \([2]\) \(28311552\) \(3.0963\)  
286650.ho3 286650ho2 \([1, -1, 0, -5441067, 4769109841]\) \(13293525831769/365192100\) \(489392341200689062500\) \([2, 2]\) \(14155776\) \(2.7497\)  
286650.ho4 286650ho1 \([1, -1, 0, 71433, 243347341]\) \(30080231/19110000\) \(-25609227692343750000\) \([2]\) \(7077888\) \(2.4032\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 286650.2.a.ho

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.