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Properties

Label 14586.e
Number of curves $4$
Conductor $14586$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 14586.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
14586.e1 14586h3 \([1, 0, 1, -3295221, -2302254848]\) \(3957101249824708884951625/772310238681366528\) \(772310238681366528\) \([2]\) \(456192\) \(2.4321\)  
14586.e2 14586h4 \([1, 0, 1, -2947061, -2807643904]\) \(-2830680648734534916567625/1766676274677722124288\) \(-1766676274677722124288\) \([2]\) \(912384\) \(2.7786\)  
14586.e3 14586h1 \([1, 0, 1, -100326, 7911736]\) \(111675519439697265625/37528570137307392\) \(37528570137307392\) \([6]\) \(152064\) \(1.8828\) \(\Gamma_0(N)\)-optimal
14586.e4 14586h2 \([1, 0, 1, 292714, 54762104]\) \(2773679829880629422375/2899504554614368272\) \(-2899504554614368272\) \([6]\) \(304128\) \(2.2293\)  

Rank

sage: E.rank()
 

The elliptic curves in class 14586.e have rank \(1\).

Complex multiplication

The elliptic curves in class 14586.e do not have complex multiplication.

Modular form 14586.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{11} + q^{12} + q^{13} + 4 q^{14} + q^{16} + q^{17} - q^{18} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.