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Properties

Label 11094q
Number of curves $2$
Conductor $11094$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -2456435, 1482324321]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -2456435, 1482324321]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -2456435, 1482324321]) 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 11094q have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(43\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + T + 5 T^{2}\) 1.5.b
\(7\) \( 1 - 5 T + 7 T^{2}\) 1.7.af
\(11\) \( 1 - T + 11 T^{2}\) 1.11.ab
\(13\) \( 1 + 3 T + 13 T^{2}\) 1.13.d
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 7 T + 19 T^{2}\) 1.19.ah
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 - 3 T + 29 T^{2}\) 1.29.ad
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 11094q do not have complex multiplication.

Modular form 11094.2.a.q

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);
 
\(q + q^{2} + q^{3} + q^{4} - 4 q^{5} + q^{6} + 4 q^{7} + q^{8} + q^{9} - 4 q^{10} - q^{11} + q^{12} - 4 q^{13} + 4 q^{14} - 4 q^{15} + q^{16} - 5 q^{17} + q^{18} + 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)\)

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 11094q

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
11094.o2 11094q1 \([1, 0, 0, -2456435, 1482324321]\) \(-140246460241/73728\) \(-861747630066966528\) \([]\) \(563472\) \(2.3918\) \(\Gamma_0(N)\)-optimal
11094.o1 11094q2 \([1, 0, 0, -151134525, -805603568589]\) \(-32663831300214001/5083731656658\) \(-59419673760599027219917458\) \([]\) \(7325136\) \(3.6743\)