Properties

Label 55440.a
Number of curves $4$
Conductor $55440$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 55440.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
55440.a1 55440j4 \([0, 0, 0, -40683, 3157562]\) \(4987755354962/1537305\) \(2295184066560\) \([2]\) \(131072\) \(1.3485\)  
55440.a2 55440j2 \([0, 0, 0, -2883, 35282]\) \(3550014724/1334025\) \(995844326400\) \([2, 2]\) \(65536\) \(1.0019\)  
55440.a3 55440j1 \([0, 0, 0, -1263, -16882]\) \(1193895376/31185\) \(5819869440\) \([2]\) \(32768\) \(0.65532\) \(\Gamma_0(N)\)-optimal
55440.a4 55440j3 \([0, 0, 0, 8997, 251498]\) \(53946017998/49520625\) \(-73933896960000\) \([2]\) \(131072\) \(1.3485\)  

Rank

sage: E.rank()
 

The elliptic curves in class 55440.a have rank \(2\).

Complex multiplication

The elliptic curves in class 55440.a do not have complex multiplication.

Modular form 55440.2.a.a

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