Properties

Label 54720bc
Number of curves $4$
Conductor $54720$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bc1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 54720bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54720.bj3 54720bc1 \([0, 0, 0, -17868, -913808]\) \(3301293169/22800\) \(4357147852800\) \([2]\) \(98304\) \(1.2588\) \(\Gamma_0(N)\)-optimal
54720.bj2 54720bc2 \([0, 0, 0, -29388, 408688]\) \(14688124849/8122500\) \(1552233922560000\) \([2, 2]\) \(196608\) \(1.6054\)  
54720.bj4 54720bc3 \([0, 0, 0, 114612, 3231088]\) \(871257511151/527800050\) \(-100864160287948800\) \([2]\) \(393216\) \(1.9519\)  
54720.bj1 54720bc4 \([0, 0, 0, -357708, 82226032]\) \(26487576322129/44531250\) \(8510054400000000\) \([2]\) \(393216\) \(1.9519\)  

Rank

sage: E.rank()
 

The elliptic curves in class 54720bc have rank \(1\).

Complex multiplication

The elliptic curves in class 54720bc do not have complex multiplication.

Modular form 54720.2.a.bc

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