Properties

Label 25857h
Number of curves $6$
Conductor $25857$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("25857.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 25857h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25857.f4 25857h1 [1, -1, 1, -819851, -285520998] [2] 172032 \(\Gamma_0(N)\)-optimal
25857.f3 25857h2 [1, -1, 1, -827456, -279948054] [2, 2] 344064  
25857.f5 25857h3 [1, -1, 1, 336109, -1005547188] [2] 688128  
25857.f2 25857h4 [1, -1, 1, -2112701, 802228236] [2, 2] 688128  
25857.f6 25857h5 [1, -1, 1, 5895364, 5427686580] [2] 1376256  
25857.f1 25857h6 [1, -1, 1, -30684686, 65420629512] [2] 1376256  

Rank

sage: E.rank()
 

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

Modular form 25857.2.a.f

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.