Properties

Label 6720.ca
Number of curves $4$
Conductor $6720$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ca1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6720.ca

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.ca1 6720ci3 \([0, 1, 0, -89185, -6377665]\) \(2394165105226952/854262178245\) \(27992463056732160\) \([4]\) \(61440\) \(1.8568\)  
6720.ca2 6720ci2 \([0, 1, 0, -79385, -8633625]\) \(13507798771700416/3544416225\) \(14517928857600\) \([2, 2]\) \(30720\) \(1.5103\)  
6720.ca3 6720ci1 \([0, 1, 0, -79380, -8634762]\) \(864335783029582144/59535\) \(3810240\) \([2]\) \(15360\) \(1.1637\) \(\Gamma_0(N)\)-optimal
6720.ca4 6720ci4 \([0, 1, 0, -69665, -10816737]\) \(-1141100604753992/875529151875\) \(-28689339248640000\) \([2]\) \(61440\) \(1.8568\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 6720.2.a.ca

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.