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Properties

Label 335730u
Number of curves $3$
Conductor $335730$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 335730u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
335730.u3 335730u1 \([1, 1, 0, -530415502, -4571267574476]\) \(350792849898814825511281/11141148807000000000\) \(524145160977413967000000000\) \([]\) \(153964800\) \(3.9015\) \(\Gamma_0(N)\)-optimal
335730.u2 335730u2 \([1, 1, 0, -5842530502, 170329007642524]\) \(468818856965932972707671281/4896432946801144503000\) \(230357001739685974951942143000\) \([]\) \(461894400\) \(4.4508\)  
335730.u1 335730u3 \([1, 1, 0, -472052316352, 124833925910650354]\) \(247270613043280364880287393857681/288395676136025670\) \(13567828660410003483765270\) \([]\) \(1385683200\) \(5.0001\)  

Rank

sage: E.rank()
 

The elliptic curves in class 335730u have rank \(1\).

Complex multiplication

The elliptic curves in class 335730u do not have complex multiplication.

Modular form 335730.2.a.u

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + 3 q^{11} - q^{12} - 5 q^{13} + q^{14} - q^{15} + q^{16} - q^{18} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.