Properties

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

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Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 5040m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5040.l4 5040m1 \([0, 0, 0, -1578, 24127]\) \(37256083456/525\) \(6123600\) \([2]\) \(2048\) \(0.44250\) \(\Gamma_0(N)\)-optimal
5040.l3 5040m2 \([0, 0, 0, -1623, 22678]\) \(2533446736/275625\) \(51438240000\) \([2, 2]\) \(4096\) \(0.78907\)  
5040.l2 5040m3 \([0, 0, 0, -6123, -160022]\) \(34008619684/4862025\) \(3629482214400\) \([2, 2]\) \(8192\) \(1.1356\)  
5040.l5 5040m4 \([0, 0, 0, 2157, 112642]\) \(1486779836/8203125\) \(-6123600000000\) \([2]\) \(8192\) \(1.1356\)  
5040.l1 5040m5 \([0, 0, 0, -94323, -11149742]\) \(62161150998242/1607445\) \(2399902525440\) \([2]\) \(16384\) \(1.4822\)  
5040.l6 5040m6 \([0, 0, 0, 10077, -863102]\) \(75798394558/259416045\) \(-387306079856640\) \([2]\) \(16384\) \(1.4822\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 5040.2.a.m

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.