Properties

Label 55440dz
Number of curves $8$
Conductor $55440$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("dz1")
 
E.isogeny_class()
 

Elliptic curves in class 55440dz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
55440.cx8 55440dz1 \([0, 0, 0, 228813, 5274866]\) \(443688652450511/260789760000\) \(-778714050723840000\) \([2]\) \(663552\) \(2.1224\) \(\Gamma_0(N)\)-optimal
55440.cx7 55440dz2 \([0, 0, 0, -923187, 42369266]\) \(29141055407581489/16604321025600\) \(49580236913305190400\) \([2, 2]\) \(1327104\) \(2.4690\)  
55440.cx6 55440dz3 \([0, 0, 0, -2916147, -2104412686]\) \(-918468938249433649/109183593750000\) \(-326020464000000000000\) \([2]\) \(1990656\) \(2.6717\)  
55440.cx5 55440dz4 \([0, 0, 0, -9476787, -11178243214]\) \(31522423139920199089/164434491947880\) \(490998762004498513920\) \([2]\) \(2654208\) \(2.8156\)  
55440.cx4 55440dz5 \([0, 0, 0, -10801587, 13637023346]\) \(46676570542430835889/106752955783320\) \(318762617921700986880\) \([2]\) \(2654208\) \(2.8156\)  
55440.cx3 55440dz6 \([0, 0, 0, -47916147, -127663412686]\) \(4074571110566294433649/48828650062500\) \(145801567828224000000\) \([2, 2]\) \(3981312\) \(3.0183\)  
55440.cx1 55440dz7 \([0, 0, 0, -766656147, -8170507760686]\) \(16689299266861680229173649/2396798250\) \(7156801225728000\) \([2]\) \(7962624\) \(3.3649\)  
55440.cx2 55440dz8 \([0, 0, 0, -49176147, -120595064686]\) \(4404531606962679693649/444872222400201750\) \(1328381338131444022272000\) \([2]\) \(7962624\) \(3.3649\)  

Rank

sage: E.rank()
 

The elliptic curves in class 55440dz have rank \(0\).

Complex multiplication

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

Modular form 55440.2.a.dz

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.