Properties

Label 223440dc
Number of curves $4$
Conductor $223440$
CM no
Rank $2$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("dc1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 223440dc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
223440.ch3 223440dc1 \([0, -1, 0, -24320, 1459200]\) \(3301293169/22800\) \(10987098931200\) \([2]\) \(589824\) \(1.3359\) \(\Gamma_0(N)\)-optimal
223440.ch2 223440dc2 \([0, -1, 0, -40000, -635648]\) \(14688124849/8122500\) \(3914153994240000\) \([2, 2]\) \(1179648\) \(1.6824\)  
223440.ch4 223440dc3 \([0, -1, 0, 156000, -5182848]\) \(871257511151/527800050\) \(-254341726545715200\) \([2]\) \(2359296\) \(2.0290\)  
223440.ch1 223440dc4 \([0, -1, 0, -486880, -130409600]\) \(26487576322129/44531250\) \(21459177600000000\) \([2]\) \(2359296\) \(2.0290\)  

Rank

sage: E.rank()
 

The elliptic curves in class 223440dc have rank \(2\).

Complex multiplication

The elliptic curves in class 223440dc do not have complex multiplication.

Modular form 223440.2.a.dc

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