Properties

Label 344850.f
Number of curves $8$
Conductor $344850$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 344850.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
344850.f1 344850f8 \([1, 1, 0, -64808454625, -6307398167652875]\) \(1087533321226184807035053481/8484255812957933638080\) \(234849636129055779278291295000000\) \([2]\) \(2229534720\) \(5.0399\)  
344850.f2 344850f5 \([1, 1, 0, -64686622750, -6332448672191000]\) \(1081411559614045490773061881/522522049500\) \(14463745070847960937500\) \([2]\) \(743178240\) \(4.4906\)  
344850.f3 344850f6 \([1, 1, 0, -6830094625, 53929513187125]\) \(1272998045160051207059881/691293848290254950400\) \(19135456580795817971649600000000\) \([2, 2]\) \(1114767360\) \(4.6933\)  
344850.f4 344850f3 \([1, 1, 0, -5281294625, 147537436387125]\) \(588530213343917460371881/861551575695360000\) \(23848299546725744640000000000\) \([2]\) \(557383680\) \(4.3468\)  
344850.f5 344850f2 \([1, 1, 0, -4042935250, -98944677753500]\) \(264020672568758737421881/5803468580250000\) \(160643728148379222656250000\) \([2, 2]\) \(371589120\) \(4.1440\)  
344850.f6 344850f4 \([1, 1, 0, -3899247750, -106303058316000]\) \(-236859095231405581781881/39282983014374049500\) \(-1087378135498867273535460937500\) \([2]\) \(743178240\) \(4.4906\)  
344850.f7 344850f1 \([1, 1, 0, -261685250, -1430021503500]\) \(71595431380957421881/9522562500000000\) \(263590630391601562500000000\) \([2]\) \(185794560\) \(3.7975\) \(\Gamma_0(N)\)-optimal
344850.f8 344850f7 \([1, 1, 0, 26367465375, 424381085227125]\) \(73240740785321709623685719/45195275784938365817280\) \(-1251034186950643691962912095000000\) \([2]\) \(2229534720\) \(5.0399\)  

Rank

sage: E.rank()
 

The elliptic curves in class 344850.f have rank \(0\).

Complex multiplication

The elliptic curves in class 344850.f do not have complex multiplication.

Modular form 344850.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - 4 q^{7} - q^{8} + q^{9} - q^{12} + 2 q^{13} + 4 q^{14} + q^{16} + 6 q^{17} - q^{18} - q^{19} + O(q^{20})\) 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.