Properties

Label 444360.bm
Number of curves $6$
Conductor $444360$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bm1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 444360.bm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
444360.bm1 444360bm5 \([0, 1, 0, -752322816, 7942198173984]\) \(155324313723954725282/13018359375\) \(3946873658162400000000\) \([2]\) \(121110528\) \(3.5864\) \(\Gamma_0(N)\)-optimal*
444360.bm2 444360bm4 \([0, 1, 0, -64749776, -200351336496]\) \(198048499826486404/242568272835\) \(36770621350219545922560\) \([2]\) \(60555264\) \(3.2398\)  
444360.bm3 444360bm3 \([0, 1, 0, -47123496, 123512273280]\) \(76343005935514084/694180580625\) \(105229966724462643840000\) \([2, 2]\) \(60555264\) \(3.2398\) \(\Gamma_0(N)\)-optimal*
444360.bm4 444360bm6 \([0, 1, 0, -13796496, 294866376480]\) \(-957928673903042/123339801817575\) \(-37393852848432191383910400\) \([2]\) \(121110528\) \(3.5864\)  
444360.bm5 444360bm2 \([0, 1, 0, -5131476, -1321603776]\) \(394315384276816/208332909225\) \(7895231340820269062400\) \([2, 2]\) \(30277632\) \(2.8933\) \(\Gamma_0(N)\)-optimal*
444360.bm6 444360bm1 \([0, 1, 0, 1219169, -160705870]\) \(84611246065664/53699121315\) \(-127190354598157984560\) \([2]\) \(15138816\) \(2.5467\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 444360.bm1.

Rank

sage: E.rank()
 

The elliptic curves in class 444360.bm have rank \(0\).

Complex multiplication

The elliptic curves in class 444360.bm do not have complex multiplication.

Modular form 444360.2.a.bm

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} - q^{7} + q^{9} + 4q^{11} - 2q^{13} - q^{15} - 2q^{17} + 4q^{19} + O(q^{20})\)  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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.