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Properties

Label 3136y
Number of curves $6$
Conductor $3136$
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, -1633, 51969]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -1633, 51969]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -1633, 51969]) 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 3136y have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 6 T + 19 T^{2}\) 1.19.ag
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 3136y do not have complex multiplication.

Modular form 3136.2.a.y

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);
 
\(q + 2 q^{3} + q^{9} - 4 q^{13} - 6 q^{17} - 2 q^{19} + O(q^{20})\) 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 3136y

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
3136.z5 3136y1 \([0, -1, 0, -1633, 51969]\) \(-15625/28\) \(-863547424768\) \([2]\) \(3072\) \(0.98059\) \(\Gamma_0(N)\)-optimal
3136.z4 3136y2 \([0, -1, 0, -32993, 2316161]\) \(128787625/98\) \(3022415986688\) \([2]\) \(6144\) \(1.3272\)  
3136.z6 3136y3 \([0, -1, 0, 14047, -1080127]\) \(9938375/21952\) \(-677021181018112\) \([2]\) \(9216\) \(1.5299\)  
3136.z3 3136y4 \([0, -1, 0, -111393, -11692351]\) \(4956477625/941192\) \(29027283136151552\) \([2]\) \(18432\) \(1.8765\)  
3136.z2 3136y5 \([0, -1, 0, -534753, -150770815]\) \(-548347731625/1835008\) \(-56593444029595648\) \([2]\) \(27648\) \(2.0792\)  
3136.z1 3136y6 \([0, -1, 0, -8562913, -9641661567]\) \(2251439055699625/25088\) \(773738492592128\) \([2]\) \(55296\) \(2.4258\)