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Properties

Label 11858bk
Number of curves $6$
Conductor $11858$
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, 1, -3088, 132397]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -3088, 132397]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -3088, 132397]) 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 11858bk have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(7\)\(1\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - T + 3 T^{2}\) 1.3.ab
\(5\) \( 1 - 4 T + 5 T^{2}\) 1.5.ae
\(13\) \( 1 - 3 T + 13 T^{2}\) 1.13.ad
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 5 T + 29 T^{2}\) 1.29.af
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 11858bk do not have complex multiplication.

Modular form 11858.2.a.bk

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);
 
\(q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{8} + q^{9} + 2 q^{12} - 4 q^{13} + q^{16} + 6 q^{17} + q^{18} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

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 11858bk

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
11858.bm5 11858bk1 \([1, 1, 1, -3088, 132397]\) \(-15625/28\) \(-5835826642492\) \([2]\) \(23040\) \(1.1398\) \(\Gamma_0(N)\)-optimal
11858.bm4 11858bk2 \([1, 1, 1, -62378, 5966533]\) \(128787625/98\) \(20425393248722\) \([2]\) \(46080\) \(1.4864\)  
11858.bm6 11858bk3 \([1, 1, 1, 26557, -2784671]\) \(9938375/21952\) \(-4575288087713728\) \([2]\) \(69120\) \(1.6891\)  
11858.bm3 11858bk4 \([1, 1, 1, -210603, -30579823]\) \(4956477625/941192\) \(196165476760726088\) \([2]\) \(138240\) \(2.0357\)  
11858.bm2 11858bk5 \([1, 1, 1, -1011018, -392829865]\) \(-548347731625/1835008\) \(-382456734842355712\) \([2]\) \(207360\) \(2.2384\)  
11858.bm1 11858bk6 \([1, 1, 1, -16189258, -25078719401]\) \(2251439055699625/25088\) \(5228900671672832\) \([2]\) \(414720\) \(2.5850\)