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Properties

Label 29744.z
Number of curves $2$
Conductor $29744$
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([0, 1, 0, -75522776, 252654335636]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -75522776, 252654335636]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -75522776, 252654335636]) 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 29744.z have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(11\)\(1 - T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - T + 3 T^{2}\) 1.3.ab
\(5\) \( 1 + T + 5 T^{2}\) 1.5.b
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(17\) \( 1 - 3 T + 17 T^{2}\) 1.17.ad
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 29744.z do not have complex multiplication.

Modular form 29744.2.a.z

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);
 
\(q + q^{3} - q^{5} + 3 q^{7} - 2 q^{9} + q^{11} - q^{15} + 3 q^{17} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 29744.z

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
29744.z1 29744ba2 \([0, 1, 0, -75522776, 252654335636]\) \(-2409558590804994721/674373039626\) \(-13332766134370850545664\) \([]\) \(2419200\) \(3.2269\)  
29744.z2 29744ba1 \([0, 1, 0, 757064, -38628524]\) \(2427173723519/1437646496\) \(-28423147707233337344\) \([]\) \(483840\) \(2.4222\) \(\Gamma_0(N)\)-optimal