Properties

Label 357390.e
Number of curves $8$
Conductor $357390$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 357390.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
357390.e1 357390e7 \([1, -1, 0, -69607493910, 7020789681355060]\) \(1087533321226184807035053481/8484255812957933638080\) \(290979831936133381674453377641920\) \([2]\) \(2229534720\) \(5.0578\)  
357390.e2 357390e4 \([1, -1, 0, -69476640435, 7048673570432425]\) \(1081411559614045490773061881/522522049500\) \(17920649907116116825500\) \([2]\) \(743178240\) \(4.5085\)  
357390.e3 357390e6 \([1, -1, 0, -7335860310, -60030595013900]\) \(1272998045160051207059881/691293848290254950400\) \(23708923001444937747154711449600\) \([2, 2]\) \(1114767360\) \(4.7112\)  
357390.e4 357390e3 \([1, -1, 0, -5672372310, -164225827474700]\) \(588530213343917460371881/861551575695360000\) \(29548158168128744734064640000\) \([2]\) \(557383680\) \(4.3646\)  
357390.e5 357390e2 \([1, -1, 0, -4342312935, 110135118570925]\) \(264020672568758737421881/5803468580250000\) \(199038354023773048232250000\) \([2, 2]\) \(371589120\) \(4.1619\)  
357390.e6 357390e5 \([1, -1, 0, -4187985435, 118325803709425]\) \(-236859095231405581781881/39282983014374049500\) \(-1347266754735842597431264825500\) \([2]\) \(743178240\) \(4.5085\)  
357390.e7 357390e1 \([1, -1, 0, -281062935, 1591714320925]\) \(71595431380957421881/9522562500000000\) \(326590062456555562500000000\) \([2]\) \(185794560\) \(3.8153\) \(\Gamma_0(N)\)-optimal
357390.e8 357390e8 \([1, -1, 0, 28319965290, -472375955747660]\) \(73240740785321709623685719/45195275784938365817280\) \(-1550037391862145731968608292662720\) \([2]\) \(2229534720\) \(5.0578\)  

Rank

sage: E.rank()
 

The elliptic curves in class 357390.e have rank \(0\).

Complex multiplication

The elliptic curves in class 357390.e do not have complex multiplication.

Modular form 357390.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - 4 q^{7} - q^{8} + q^{10} + q^{11} - 2 q^{13} + 4 q^{14} + q^{16} - 6 q^{17} + O(q^{20})\) 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.