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Properties

Label 64680bl
Number of curves $6$
Conductor $64680$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bl1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 64680bl have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(7\)\(1\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 64680bl do not have complex multiplication.

Modular form 64680.2.a.bl

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} + q^{11} + 2 q^{13} + q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 64680bl

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
64680.j5 64680bl1 \([0, -1, 0, -3156351, -2118320784]\) \(1847444944806639616/38285567941005\) \(72068140523060755920\) \([2]\) \(1769472\) \(2.6005\) \(\Gamma_0(N)\)-optimal
64680.j4 64680bl2 \([0, -1, 0, -6743396, 3589385220]\) \(1125982298608534096/467044181552025\) \(14066503914346032441600\) \([2, 2]\) \(3538944\) \(2.9471\)  
64680.j6 64680bl3 \([0, -1, 0, 22308704, 26261644060]\) \(10191978981888338876/8372623608979245\) \(-1008671534052146375685120\) \([2]\) \(7077888\) \(3.2937\)  
64680.j2 64680bl4 \([0, -1, 0, -93188216, 346152917916]\) \(742879737792994384804/317817082130625\) \(38288242581079962240000\) \([2, 2]\) \(7077888\) \(3.2937\)  
64680.j3 64680bl5 \([0, -1, 0, -78635216, 457914136716]\) \(-223180773010681046402/246754509479287425\) \(-59454302793172349468313600\) \([2]\) \(14155776\) \(3.6402\)  
64680.j1 64680bl6 \([0, -1, 0, -1490858336, 22157074674540]\) \(1520949008089505953959842/278553515625\) \(67116119162400000000\) \([2]\) \(14155776\) \(3.6402\)