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Properties

Label 25857j
Number of curves $2$
Conductor $25857$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 25857j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
25857.j2 25857j1 \([1, -1, 1, -1114418, -451797056]\) \(43499078731809/82055753\) \(288733168922907033\) \([2]\) \(645120\) \(2.2404\) \(\Gamma_0(N)\)-optimal
25857.j1 25857j2 \([1, -1, 1, -17822603, -28955960666]\) \(177930109857804849/634933\) \(2234166532403013\) \([2]\) \(1290240\) \(2.5870\)  

Rank

sage: E.rank()
 

The elliptic curves in class 25857j have rank \(0\).

Complex multiplication

The elliptic curves in class 25857j do not have complex multiplication.

Modular form 25857.2.a.j

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