Properties

Label 135240do
Number of curves $4$
Conductor $135240$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("do1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 135240do

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
135240.c4 135240do1 \([0, -1, 0, -49996, 295156]\) \(458891455696/264449745\) \(7964735500673280\) \([2]\) \(737280\) \(1.7407\) \(\Gamma_0(N)\)-optimal
135240.c2 135240do2 \([0, -1, 0, -568416, 164737980]\) \(168591300897604/472410225\) \(56912476734489600\) \([2, 2]\) \(1474560\) \(2.0873\)  
135240.c1 135240do3 \([0, -1, 0, -9088536, 10549060236]\) \(344577854816148242/2716875\) \(654617859840000\) \([2]\) \(2949120\) \(2.4339\)  
135240.c3 135240do4 \([0, -1, 0, -343016, 296461740]\) \(-18524646126002/146738831715\) \(-35356010111873095680\) \([2]\) \(2949120\) \(2.4339\)  

Rank

sage: E.rank()
 

The elliptic curves in class 135240do have rank \(0\).

Complex multiplication

The elliptic curves in class 135240do do not have complex multiplication.

Modular form 135240.2.a.do

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} - 4 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\)  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.