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Properties

Label 10890ba
Number of curves $8$
Conductor $10890$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 0, 1611, 76005]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, 1611, 76005]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, 1611, 76005]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 10890ba have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 10890ba do not have complex multiplication.

Modular form 10890.2.a.ba

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{2} + q^{4} + q^{5} + 4 q^{7} - q^{8} - q^{10} - 2 q^{13} - 4 q^{14} + q^{16} + 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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 comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 10890ba

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10890.ba8 10890ba1 \([1, -1, 0, 1611, 76005]\) \(357911/2160\) \(-2789570813040\) \([2]\) \(23040\) \(1.0700\) \(\Gamma_0(N)\)-optimal
10890.ba6 10890ba2 \([1, -1, 0, -20169, 1003833]\) \(702595369/72900\) \(94148014940100\) \([2, 2]\) \(46080\) \(1.4165\)  
10890.ba7 10890ba3 \([1, -1, 0, -14724, -2246832]\) \(-273359449/1536000\) \(-1983694800384000\) \([2]\) \(69120\) \(1.6193\)  
10890.ba5 10890ba4 \([1, -1, 0, -74619, -6738957]\) \(35578826569/5314410\) \(6863390289133290\) \([2]\) \(92160\) \(1.7631\)  
10890.ba4 10890ba5 \([1, -1, 0, -314199, 67866255]\) \(2656166199049/33750\) \(43587043953750\) \([2]\) \(92160\) \(1.7631\)  
10890.ba3 10890ba6 \([1, -1, 0, -363204, -84000240]\) \(4102915888729/9000000\) \(11623211721000000\) \([2, 2]\) \(138240\) \(1.9658\)  
10890.ba1 10890ba7 \([1, -1, 0, -5808204, -5386341240]\) \(16778985534208729/81000\) \(104608905489000\) \([2]\) \(276480\) \(2.3124\)  
10890.ba2 10890ba8 \([1, -1, 0, -493884, -18059112]\) \(10316097499609/5859375000\) \(7567195130859375000\) \([2]\) \(276480\) \(2.3124\)