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Properties

Label 80850bz
Number of curves $8$
Conductor $80850$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bz1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 80850bz have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1\)
\(11\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 80850bz do not have complex multiplication.

Modular form 80850.2.a.bz

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} - q^{11} + q^{12} + 2 q^{13} + q^{16} + 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 80850bz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
80850.cl8 80850bz1 \([1, 0, 1, 1946499, -130717352]\) \(443688652450511/260789760000\) \(-479400851160000000000\) \([2]\) \(3981312\) \(2.6576\) \(\Gamma_0(N)\)-optimal
80850.cl7 80850bz2 \([1, 0, 1, -7853501, -1051917352]\) \(29141055407581489/16604321025600\) \(30523152567825225000000\) \([2, 2]\) \(7962624\) \(3.0042\)  
80850.cl6 80850bz3 \([1, 0, 1, -24807501, 52212454648]\) \(-918468938249433649/109183593750000\) \(-200708447204589843750000\) \([2]\) \(11943936\) \(3.2069\)  
80850.cl5 80850bz4 \([1, 0, 1, -80618501, 277346972648]\) \(31522423139920199089/164434491947880\) \(302274274112127095625000\) \([2]\) \(15925248\) \(3.3508\)  
80850.cl4 80850bz5 \([1, 0, 1, -91888501, -338368407352]\) \(46676570542430835889/106752955783320\) \(196240288983622104375000\) \([2]\) \(15925248\) \(3.3508\)  
80850.cl3 80850bz6 \([1, 0, 1, -407620001, 3167540579648]\) \(4074571110566294433649/48828650062500\) \(89760028925047851562500\) \([2, 2]\) \(23887872\) \(3.5535\)  
80850.cl1 80850bz7 \([1, 0, 1, -6521901251, 202725452017148]\) \(16689299266861680229173649/2396798250\) \(4405951833035156250\) \([2]\) \(47775744\) \(3.9001\)  
80850.cl2 80850bz8 \([1, 0, 1, -418338751, 2992160392148]\) \(4404531606962679693649/444872222400201750\) \(817793313955645870089843750\) \([2]\) \(47775744\) \(3.9001\)