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Properties

Label 178186.d
Number of curves $2$
Conductor $178186$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 178186.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
178186.d1 178186a2 \([1, 1, 1, -41357678, -102389504853]\) \(-1646982616152408625/38112512\) \(-181038404886363392\) \([]\) \(9953280\) \(2.8342\)  
178186.d2 178186a1 \([1, 1, 1, -475758, -160606037]\) \(-2507141976625/889192448\) \(-4223756818309971968\) \([]\) \(3317760\) \(2.2849\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 178186.d have rank \(1\).

Complex multiplication

The elliptic curves in class 178186.d do not have complex multiplication.

Modular form 178186.2.a.d

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} + 4 q^{7} + q^{8} - 2 q^{9} - q^{12} - 5 q^{13} + 4 q^{14} + q^{16} + 3 q^{17} - 2 q^{18} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.