Properties

Label 6720.m
Number of curves $6$
Conductor $6720$
CM no
Rank $1$
Graph

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

Elliptic curves in class 6720.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.m1 6720bm5 \([0, -1, 0, -41921, 3317601]\) \(62161150998242/1607445\) \(210691031040\) \([2]\) \(16384\) \(1.2795\)  
6720.m2 6720bm3 \([0, -1, 0, -2721, 48321]\) \(34008619684/4862025\) \(318637670400\) \([2, 2]\) \(8192\) \(0.93291\)  
6720.m3 6720bm2 \([0, -1, 0, -721, -6479]\) \(2533446736/275625\) \(4515840000\) \([2, 2]\) \(4096\) \(0.58634\)  
6720.m4 6720bm1 \([0, -1, 0, -701, -6915]\) \(37256083456/525\) \(537600\) \([2]\) \(2048\) \(0.23977\) \(\Gamma_0(N)\)-optimal
6720.m5 6720bm4 \([0, -1, 0, 959, -33695]\) \(1486779836/8203125\) \(-537600000000\) \([2]\) \(8192\) \(0.93291\)  
6720.m6 6720bm6 \([0, -1, 0, 4479, 254241]\) \(75798394558/259416045\) \(-34002179850240\) \([2]\) \(16384\) \(1.2795\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6720.m have rank \(1\).

Complex multiplication

The elliptic curves in class 6720.m do not have complex multiplication.

Modular form 6720.2.a.m

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{7} + q^{9} - 4 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.