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Properties

Label 316239.r
Number of curves $2$
Conductor $316239$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 316239.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
316239.r1 316239r2 \([1, 0, 1, -25556521, 28465194035]\) \(719479696477074625/279917719389693\) \(718192284985186692502437\) \([2]\) \(31518720\) \(3.2764\)  
316239.r2 316239r1 \([1, 0, 1, 5088544, 3201402449]\) \(5679290619623375/5011824295863\) \(-12858969953163528545967\) \([2]\) \(15759360\) \(2.9298\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 316239.r have rank \(1\).

Complex multiplication

The elliptic curves in class 316239.r do not have complex multiplication.

Modular form 316239.2.a.r

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} + q^{6} + q^{7} - 3 q^{8} + q^{9} - q^{11} - q^{12} + q^{14} - q^{16} + q^{18} - 6 q^{19} + O(q^{20})\) 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.