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Properties

Label 68970bg
Number of curves $4$
Conductor $68970$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bg1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 68970bg have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(11\)\(1\)
\(19\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(13\) \( 1 + T + 13 T^{2}\) 1.13.b
\(17\) \( 1 - 5 T + 17 T^{2}\) 1.17.af
\(23\) \( 1 - T + 23 T^{2}\) 1.23.ab
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 68970bg do not have complex multiplication.

Modular form 68970.2.a.bg

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

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 68970bg

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68970.bh4 68970bg1 \([1, 0, 1, 1129532, 982586306]\) \(89962967236397039/287450726400000\) \(-509236496311910400000\) \([2]\) \(3360000\) \(2.6565\) \(\Gamma_0(N)\)-optimal
68970.bh3 68970bg2 \([1, 0, 1, -10641348, 11510461378]\) \(75224183150104868881/11219310000000000\) \(19875692042910000000000\) \([2]\) \(6720000\) \(3.0031\)  
68970.bh2 68970bg3 \([1, 0, 1, -399477268, 3073135105346]\) \(-3979640234041473454886161/1471455901872240\) \(-2606773888976687366640\) \([2]\) \(16800000\) \(3.4612\)  
68970.bh1 68970bg4 \([1, 0, 1, -6391636848, 196682207998978]\) \(16300610738133468173382620881/2228489100\) \(3947904378485100\) \([2]\) \(33600000\) \(3.8078\)