Properties

Label 5040.i
Number of curves $6$
Conductor $5040$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 5040.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5040.i1 5040bg5 \([0, 0, 0, -2419203, -1448293502]\) \(524388516989299201/3150\) \(9405849600\) \([2]\) \(49152\) \(1.9792\)  
5040.i2 5040bg3 \([0, 0, 0, -151203, -22628702]\) \(128031684631201/9922500\) \(29628426240000\) \([2, 2]\) \(24576\) \(1.6326\)  
5040.i3 5040bg6 \([0, 0, 0, -141123, -25775678]\) \(-104094944089921/35880468750\) \(-107138505600000000\) \([2]\) \(49152\) \(1.9792\)  
5040.i4 5040bg4 \([0, 0, 0, -53283, 4474402]\) \(5602762882081/345888060\) \(1032816212951040\) \([2]\) \(24576\) \(1.6326\)  
5040.i5 5040bg2 \([0, 0, 0, -10083, -303518]\) \(37966934881/8643600\) \(25809651302400\) \([2, 2]\) \(12288\) \(1.2861\)  
5040.i6 5040bg1 \([0, 0, 0, 1437, -29342]\) \(109902239/188160\) \(-561842749440\) \([2]\) \(6144\) \(0.93949\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 5040.i have rank \(1\).

Complex multiplication

The elliptic curves in class 5040.i do not have complex multiplication.

Modular form 5040.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{5} - q^{7} + 4 q^{11} - 2 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.