Properties

Label 129360.j
Number of curves $4$
Conductor $129360$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 129360.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129360.j1 129360k4 \([0, -1, 0, -27474094216, 1752805928295136]\) \(19037313645387618625546168804/82399233032965368135\) \(9926847863905630818011765760\) \([2]\) \(181665792\) \(4.5790\)  
129360.j2 129360k2 \([0, -1, 0, -1744257916, 26477999648416]\) \(19486220601593009351102416/1221175284018082695225\) \(36779533053297513218694662400\) \([2, 2]\) \(90832896\) \(4.2325\)  
129360.j3 129360k1 \([0, -1, 0, -331881671, -1818675944910]\) \(2147658844706816042407936/483688189481299210485\) \(910486908868565933029596240\) \([2]\) \(45416448\) \(3.8859\) \(\Gamma_0(N)\)-optimal
129360.j4 129360k3 \([0, -1, 0, 1387558464, 111114710954640]\) \(2452389160534358561651516/45692546768053107181875\) \(-5504698813147832327004581760000\) \([2]\) \(181665792\) \(4.5790\)  

Rank

sage: E.rank()
 

The elliptic curves in class 129360.j have rank \(0\).

Complex multiplication

The elliptic curves in class 129360.j do not have complex multiplication.

Modular form 129360.2.a.j

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