Properties

Label 319725t
Number of curves $6$
Conductor $319725$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("319725.t1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 319725t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
319725.t5 319725t1 [1, -1, 1, -8643830, 10475024172] [2] 18874368 \(\Gamma_0(N)\)-optimal
319725.t4 319725t2 [1, -1, 1, -140998955, 644456072922] [2, 2] 37748736  
319725.t1 319725t3 [1, -1, 1, -2255979830, 41243628949422] [2] 75497472  
319725.t3 319725t4 [1, -1, 1, -143700080, 618482054922] [2, 2] 75497472  
319725.t6 319725t5 [1, -1, 1, 137602795, 2744569184172] [2] 150994944  
319725.t2 319725t6 [1, -1, 1, -468220955, -3169974639828] [2] 150994944  

Rank

sage: E.rank()
 

The elliptic curves in class 319725t have rank \(1\).

Modular form 319725.2.a.t

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{4} + 3q^{8} - 4q^{11} - 2q^{13} - q^{16} - 2q^{17} + 4q^{19} + O(q^{20}) \)

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.