Properties

Label 223440.fl
Number of curves $4$
Conductor $223440$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("fl1")
 
E.isogeny_class()
 

Elliptic curves in class 223440.fl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
223440.fl1 223440b4 \([0, 1, 0, -363173120, 1700907890100]\) \(10993009831928446009969/3767761230468750000\) \(1815647604750000000000000000\) \([2]\) \(149299200\) \(3.9325\)  
223440.fl2 223440b2 \([0, 1, 0, -325352960, 2258704221108]\) \(7903870428425797297009/886464000000\) \(427178406445056000000\) \([2]\) \(49766400\) \(3.3832\)  
223440.fl3 223440b1 \([0, 1, 0, -20282880, 35475506100]\) \(-1914980734749238129/20440940544000\) \(-9850291052794085376000\) \([2]\) \(24883200\) \(3.0366\) \(\Gamma_0(N)\)-optimal
223440.fl4 223440b3 \([0, 1, 0, 67023360, 184723415988]\) \(69096190760262356111/70568821500000000\) \(-34006430845556736000000000\) \([2]\) \(74649600\) \(3.5859\)  

Rank

sage: E.rank()
 

The elliptic curves in class 223440.fl have rank \(0\).

Complex multiplication

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

Modular form 223440.2.a.fl

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{9} - 6 q^{11} + 4 q^{13} + q^{15} + 6 q^{17} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.