Properties

Label 129285n
Number of curves $2$
Conductor $129285$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("n1")
 
E.isogeny_class()
 

Elliptic curves in class 129285n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129285.g2 129285n1 \([1, -1, 1, -25382, 1661716]\) \(-19034163/1445\) \(-137283787835415\) \([2]\) \(442368\) \(1.4609\) \(\Gamma_0(N)\)-optimal
129285.g1 129285n2 \([1, -1, 1, -413237, 102348874]\) \(82142689923/425\) \(40377584657475\) \([2]\) \(884736\) \(1.8075\)  

Rank

sage: E.rank()
 

The elliptic curves in class 129285n have rank \(1\).

Complex multiplication

The elliptic curves in class 129285n do not have complex multiplication.

Modular form 129285.2.a.n

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + q^{5} - 4 q^{7} + 3 q^{8} - q^{10} - 2 q^{11} + 4 q^{14} - q^{16} + q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.