Properties

Label 164730.dj
Number of curves $4$
Conductor $164730$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("dj1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 164730.dj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
164730.dj1 164730d4 \([1, 0, 0, -179475, -29237793]\) \(26487576322129/44531250\) \(1074876119531250\) \([2]\) \(1310720\) \(1.7795\)  
164730.dj2 164730d2 \([1, 0, 0, -14745, -146475]\) \(14688124849/8122500\) \(196057404202500\) \([2, 2]\) \(655360\) \(1.4330\)  
164730.dj3 164730d1 \([1, 0, 0, -8965, 324017]\) \(3301293169/22800\) \(550336573200\) \([2]\) \(327680\) \(1.0864\) \(\Gamma_0(N)\)-optimal
164730.dj4 164730d3 \([1, 0, 0, 57505, -1143525]\) \(871257511151/527800050\) \(-12739810125078450\) \([2]\) \(1310720\) \(1.7795\)  

Rank

sage: E.rank()
 

The elliptic curves in class 164730.dj have rank \(0\).

Complex multiplication

The elliptic curves in class 164730.dj do not have complex multiplication.

Modular form 164730.2.a.dj

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} + 2 q^{13} + q^{15} + q^{16} + q^{18} - q^{19} + O(q^{20})\)  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.