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Properties

Label 6270l
Number of curves $8$
Conductor $6270$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("l1")
 
E.isogeny_class()
 

Elliptic curves in class 6270l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6270.l7 6270l1 \([1, 0, 1, -86508, 8587306]\) \(71595431380957421881/9522562500000000\) \(9522562500000000\) \([6]\) \(64512\) \(1.7938\) \(\Gamma_0(N)\)-optimal
6270.l5 6270l2 \([1, 0, 1, -1336508, 594587306]\) \(264020672568758737421881/5803468580250000\) \(5803468580250000\) \([2, 6]\) \(129024\) \(2.1404\)  
6270.l4 6270l3 \([1, 0, 1, -1745883, -886935194]\) \(588530213343917460371881/861551575695360000\) \(861551575695360000\) \([2]\) \(193536\) \(2.3431\)  
6270.l2 6270l4 \([1, 0, 1, -21384008, 38059355306]\) \(1081411559614045490773061881/522522049500\) \(522522049500\) \([6]\) \(258048\) \(2.4869\)  
6270.l6 6270l5 \([1, 0, 1, -1289008, 638819306]\) \(-236859095231405581781881/39282983014374049500\) \(-39282983014374049500\) \([6]\) \(258048\) \(2.4869\)  
6270.l3 6270l6 \([1, 0, 1, -2257883, -324349594]\) \(1272998045160051207059881/691293848290254950400\) \(691293848290254950400\) \([2, 2]\) \(387072\) \(2.6897\)  
6270.l1 6270l7 \([1, 0, 1, -21424283, 37908785126]\) \(1087533321226184807035053481/8484255812957933638080\) \(8484255812957933638080\) \([2]\) \(774144\) \(3.0362\)  
6270.l8 6270l8 \([1, 0, 1, 8716517, -2549957914]\) \(73240740785321709623685719/45195275784938365817280\) \(-45195275784938365817280\) \([2]\) \(774144\) \(3.0362\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6270l have rank \(0\).

Complex multiplication

The elliptic curves in class 6270l do not have complex multiplication.

Modular form 6270.2.a.l

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} - q^{10} - q^{11} + q^{12} + 2 q^{13} + 4 q^{14} + q^{15} + q^{16} + 6 q^{17} - q^{18} + q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.