Properties

Label 55488cp
Number of curves $6$
Conductor $55488$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cp1")
 
E.isogeny_class()
 

Elliptic curves in class 55488cp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
55488.m5 55488cp1 \([0, -1, 0, -629249, -177971295]\) \(4354703137/352512\) \(2230526338224095232\) \([2]\) \(884736\) \(2.2637\) \(\Gamma_0(N)\)-optimal
55488.m4 55488cp2 \([0, -1, 0, -2108929, 972923809]\) \(163936758817/30338064\) \(191964672983411195904\) \([2, 2]\) \(1769472\) \(2.6103\)  
55488.m6 55488cp3 \([0, -1, 0, 4179711, 5660476065]\) \(1276229915423/2927177028\) \(-18521767933002365534208\) \([2]\) \(3538944\) \(2.9568\)  
55488.m2 55488cp4 \([0, -1, 0, -32072449, 69918983329]\) \(576615941610337/27060804\) \(171227748432734060544\) \([2, 2]\) \(3538944\) \(2.9568\)  
55488.m3 55488cp5 \([0, -1, 0, -30407809, 77498755105]\) \(-491411892194497/125563633938\) \(-794506265380576748371968\) \([2]\) \(7077888\) \(3.3034\)  
55488.m1 55488cp6 \([0, -1, 0, -513153409, 4474407604513]\) \(2361739090258884097/5202\) \(32915753255043072\) \([2]\) \(7077888\) \(3.3034\)  

Rank

sage: E.rank()
 

The elliptic curves in class 55488cp have rank \(0\).

Complex multiplication

The elliptic curves in class 55488cp do not have complex multiplication.

Modular form 55488.2.a.cp

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.