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Properties

Label 32718.l
Number of curves $4$
Conductor $32718$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("l1")
 
E.isogeny_class()
 

Elliptic curves in class 32718.l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32718.l1 32718w4 \([1, 0, 0, -141574640, -612508800804]\) \(313819633022945271834814191361/19542669977582921378655012\) \(19542669977582921378655012\) \([2]\) \(19200000\) \(3.6043\)  
32718.l2 32718w2 \([1, 0, 0, -25200620, 48690669456]\) \(1769935296778757928512038081/2244956575342666752\) \(2244956575342666752\) \([10]\) \(3840000\) \(2.7996\)  
32718.l3 32718w1 \([1, 0, 0, -1561580, 774335376]\) \(-421130255542777411888321/15402588724729479168\) \(-15402588724729479168\) \([10]\) \(1920000\) \(2.4530\) \(\Gamma_0(N)\)-optimal
32718.l4 32718w3 \([1, 0, 0, 6991300, -40084233984]\) \(37791795265406275661467199/716004621065775357793008\) \(-716004621065775357793008\) \([2]\) \(9600000\) \(3.2577\)  

Rank

sage: E.rank()
 

The elliptic curves in class 32718.l have rank \(1\).

Complex multiplication

The elliptic curves in class 32718.l do not have complex multiplication.

Modular form 32718.2.a.l

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - 4 q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - 4 q^{10} + 2 q^{11} + q^{12} - 6 q^{13} + q^{14} - 4 q^{15} + q^{16} - 2 q^{17} + q^{18} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 5 & 10 & 2 \\ 5 & 1 & 2 & 10 \\ 10 & 2 & 1 & 5 \\ 2 & 10 & 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.