Properties

Label 6525.e
Number of curves $4$
Conductor $6525$
CM no
Rank $0$
Graph

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

Elliptic curves in class 6525.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6525.e1 6525f3 \([1, -1, 1, -1566005, 754680372]\) \(37286818682653441/1305\) \(14864765625\) \([2]\) \(61440\) \(1.8981\)  
6525.e2 6525f2 \([1, -1, 1, -97880, 11809122]\) \(9104453457841/1703025\) \(19398519140625\) \([2, 2]\) \(30720\) \(1.5516\)  
6525.e3 6525f4 \([1, -1, 1, -87755, 14340372]\) \(-6561258219361/3978455625\) \(-45317096103515625\) \([2]\) \(61440\) \(1.8981\)  
6525.e4 6525f1 \([1, -1, 1, -6755, 145122]\) \(2992209121/951345\) \(10836414140625\) \([2]\) \(15360\) \(1.2050\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 6525.e have rank \(0\).

Complex multiplication

The elliptic curves in class 6525.e do not have complex multiplication.

Modular form 6525.2.a.e

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