Properties

Label 88725s
Number of curves $4$
Conductor $88725$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("s1")
 
E.isogeny_class()
 

Elliptic curves in class 88725s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
88725.cb3 88725s1 \([1, 1, 0, -10650, 396375]\) \(1771561/105\) \(7918983515625\) \([2]\) \(184320\) \(1.2281\) \(\Gamma_0(N)\)-optimal
88725.cb2 88725s2 \([1, 1, 0, -31775, -1695000]\) \(47045881/11025\) \(831493269140625\) \([2, 2]\) \(368640\) \(1.5747\)  
88725.cb4 88725s3 \([1, 1, 0, 73850, -10461875]\) \(590589719/972405\) \(-73337706338203125\) \([2]\) \(737280\) \(1.9212\)  
88725.cb1 88725s4 \([1, 1, 0, -475400, -126353625]\) \(157551496201/13125\) \(989872939453125\) \([2]\) \(737280\) \(1.9212\)  

Rank

sage: E.rank()
 

The elliptic curves in class 88725s have rank \(0\).

Complex multiplication

The elliptic curves in class 88725s do not have complex multiplication.

Modular form 88725.2.a.s

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