Properties

Label 53040.j
Number of curves $4$
Conductor $53040$
CM no
Rank $2$
Graph

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

Elliptic curves in class 53040.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53040.j1 53040b4 \([0, -1, 0, -32696, 2286480]\) \(1887517194957938/21849165\) \(44747089920\) \([2]\) \(98304\) \(1.1949\)  
53040.j2 53040b2 \([0, -1, 0, -2096, 34320]\) \(994958062276/98903025\) \(101276697600\) \([2, 2]\) \(49152\) \(0.84829\)  
53040.j3 53040b1 \([0, -1, 0, -476, -3264]\) \(46689225424/7249905\) \(1855975680\) \([2]\) \(24576\) \(0.50172\) \(\Gamma_0(N)\)-optimal
53040.j4 53040b3 \([0, -1, 0, 2584, 161616]\) \(931329171502/6107473125\) \(-12508104960000\) \([2]\) \(98304\) \(1.1949\)  

Rank

sage: E.rank()
 

The elliptic curves in class 53040.j have rank \(2\).

Complex multiplication

The elliptic curves in class 53040.j do not have complex multiplication.

Modular form 53040.2.a.j

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} - 4q^{11} - q^{13} + q^{15} + q^{17} - 8q^{19} + O(q^{20})\)  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.