Properties

Label 6720.h
Number of curves $6$
Conductor $6720$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("h1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6720.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.h1 6720bh5 \([0, -1, 0, -97601, 11765985]\) \(784478485879202/221484375\) \(29030400000000\) \([2]\) \(32768\) \(1.5643\)  
6720.h2 6720bh3 \([0, -1, 0, -6881, 135681]\) \(549871953124/200930625\) \(13168189440000\) \([2, 2]\) \(16384\) \(1.2177\)  
6720.h3 6720bh2 \([0, -1, 0, -2961, -59535]\) \(175293437776/4862025\) \(79659417600\) \([2, 2]\) \(8192\) \(0.87118\)  
6720.h4 6720bh1 \([0, -1, 0, -2941, -60419]\) \(2748251600896/2205\) \(2257920\) \([2]\) \(4096\) \(0.52460\) \(\Gamma_0(N)\)-optimal
6720.h5 6720bh4 \([0, -1, 0, 639, -198495]\) \(439608956/259416045\) \(-17001089925120\) \([2]\) \(16384\) \(1.2177\)  
6720.h6 6720bh6 \([0, -1, 0, 21119, 936481]\) \(7947184069438/7533176175\) \(-987388467609600\) \([2]\) \(32768\) \(1.5643\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6720.h have rank \(0\).

Complex multiplication

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

Modular form 6720.2.a.h

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - q^{7} + q^{9} + 4q^{11} + 2q^{13} + q^{15} - 6q^{17} + 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.