Properties

Label 250800.eq
Number of curves $8$
Conductor $250800$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("eq1")
 
E.isogeny_class()
 

Elliptic curves in class 250800.eq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
250800.eq1 250800eq8 \([0, 1, 0, -8569713008, -303287420436012]\) \(1087533321226184807035053481/8484255812957933638080\) \(542992372029307752837120000000\) \([2]\) \(445906944\) \(4.5341\)  
250800.eq2 250800eq5 \([0, 1, 0, -8553603008, -304491949656012]\) \(1081411559614045490773061881/522522049500\) \(33441411168000000000\) \([2]\) \(148635648\) \(3.9848\)  
250800.eq3 250800eq6 \([0, 1, 0, -903153008, 2592990443988]\) \(1272998045160051207059881/691293848290254950400\) \(44242806290576316825600000000\) \([2, 2]\) \(222953472\) \(4.1875\)  
250800.eq4 250800eq3 \([0, 1, 0, -698353008, 7094084843988]\) \(588530213343917460371881/861551575695360000\) \(55139300844503040000000000\) \([2]\) \(111476736\) \(3.8410\)  
250800.eq5 250800eq2 \([0, 1, 0, -534603008, -4757767656012]\) \(264020672568758737421881/5803468580250000\) \(371421989136000000000000\) \([2, 2]\) \(74317824\) \(3.6382\)  
250800.eq6 250800eq4 \([0, 1, 0, -515603008, -5111585656012]\) \(-236859095231405581781881/39282983014374049500\) \(-2514110912919939168000000000\) \([4]\) \(148635648\) \(3.9848\)  
250800.eq7 250800eq1 \([0, 1, 0, -34603008, -68767656012]\) \(71595431380957421881/9522562500000000\) \(609444000000000000000000\) \([2]\) \(37158912\) \(3.2917\) \(\Gamma_0(N)\)-optimal
250800.eq8 250800eq7 \([0, 1, 0, 3486606992, 20406636523988]\) \(73240740785321709623685719/45195275784938365817280\) \(-2892497650236055412305920000000\) \([4]\) \(445906944\) \(4.5341\)  

Rank

sage: E.rank()
 

The elliptic curves in class 250800.eq have rank \(0\).

Complex multiplication

The elliptic curves in class 250800.eq do not have complex multiplication.

Modular form 250800.2.a.eq

sage: E.q_eigenform(10)
 
\(q + q^{3} - 4 q^{7} + q^{9} + q^{11} - 2 q^{13} - 6 q^{17} - q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.