Properties

Label 253704e
Number of curves $4$
Conductor $253704$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 253704e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
253704.e4 253704e1 \([0, -1, 0, 641, -171236]\) \(2048/891\) \(-12652252476336\) \([2]\) \(737280\) \(1.1932\) \(\Gamma_0(N)\)-optimal
253704.e3 253704e2 \([0, -1, 0, -42604, -3284876]\) \(37642192/1089\) \(247421826203904\) \([2, 2]\) \(1474560\) \(1.5397\)  
253704.e2 253704e3 \([0, -1, 0, -100264, 7578268]\) \(122657188/43923\) \(39917387960896512\) \([2]\) \(2949120\) \(1.8863\)  
253704.e1 253704e4 \([0, -1, 0, -676864, -214112900]\) \(37736227588/33\) \(29990524388352\) \([2]\) \(2949120\) \(1.8863\)  

Rank

sage: E.rank()
 

The elliptic curves in class 253704e have rank \(1\).

Complex multiplication

The elliptic curves in class 253704e do not have complex multiplication.

Modular form 253704.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + 4 q^{7} + q^{9} + q^{11} - 6 q^{13} + 2 q^{15} - 6 q^{17} - 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.