Properties

Label 338130.j
Number of curves $4$
Conductor $338130$
CM no
Rank $1$
Graph

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

Elliptic curves in class 338130.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
338130.j1 338130j4 \([1, -1, 0, -3110850, 2005178386]\) \(189208196468929/10860320250\) \(191101320730028270250\) \([2]\) \(11943936\) \(2.6457\)  
338130.j2 338130j2 \([1, -1, 0, -535860, -150181880]\) \(967068262369/4928040\) \(86715210134840040\) \([2]\) \(3981312\) \(2.0964\)  
338130.j3 338130j1 \([1, -1, 0, -15660, -4838000]\) \(-24137569/561600\) \(-9882075229041600\) \([2]\) \(1990656\) \(1.7499\) \(\Gamma_0(N)\)-optimal
338130.j4 338130j3 \([1, -1, 0, 140400, 127906636]\) \(17394111071/411937500\) \(-7248570806024437500\) \([2]\) \(5971968\) \(2.2992\)  

Rank

sage: E.rank()
 

The elliptic curves in class 338130.j have rank \(1\).

Complex multiplication

The elliptic curves in class 338130.j do not have complex multiplication.

Modular form 338130.2.a.j

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

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.