Properties

Label 25410h
Number of curves $4$
Conductor $25410$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("h1")
 
E.isogeny_class()
 

Elliptic curves in class 25410h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
25410.h3 25410h1 \([1, 1, 0, -60623, 4963413]\) \(13908844989649/1980372240\) \(3508350225866640\) \([2]\) \(184320\) \(1.7086\) \(\Gamma_0(N)\)-optimal
25410.h2 25410h2 \([1, 1, 0, -256643, -45178503]\) \(1055257664218129/115307784900\) \(204274774725228900\) \([2, 2]\) \(368640\) \(2.0552\)  
25410.h4 25410h3 \([1, 1, 0, 342307, -224024973]\) \(2503876820718671/13702874328990\) \(-24275477749139853390\) \([2]\) \(737280\) \(2.4017\)  
25410.h1 25410h4 \([1, 1, 0, -3991913, -3071494257]\) \(3971101377248209009/56495958750\) \(100086037179108750\) \([2]\) \(737280\) \(2.4017\)  

Rank

sage: E.rank()
 

The elliptic curves in class 25410h have rank \(1\).

Complex multiplication

The elliptic curves in class 25410h do not have complex multiplication.

Modular form 25410.2.a.h

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.