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Properties

Label 2450y
Number of curves $6$
Conductor $2450$
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, -638, -12608]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -638, -12608]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -638, -12608]) 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 2450y have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 2450y do not have complex multiplication.

Modular form 2450.2.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);
 
\(q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} + q^{8} + q^{9} - 2 q^{12} - 4 q^{13} + q^{16} + 6 q^{17} + q^{18} - 2 q^{19} + O(q^{20})\) 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 2450y

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
2450.t5 2450y1 \([1, 0, 0, -638, -12608]\) \(-15625/28\) \(-51471437500\) \([2]\) \(2304\) \(0.74559\) \(\Gamma_0(N)\)-optimal
2450.t4 2450y2 \([1, 0, 0, -12888, -563858]\) \(128787625/98\) \(180150031250\) \([2]\) \(4608\) \(1.0922\)  
2450.t6 2450y3 \([1, 0, 0, 5487, 263017]\) \(9938375/21952\) \(-40353607000000\) \([2]\) \(6912\) \(1.2949\)  
2450.t3 2450y4 \([1, 0, 0, -43513, 2860017]\) \(4956477625/941192\) \(1730160900125000\) \([2]\) \(13824\) \(1.6415\)  
2450.t2 2450y5 \([1, 0, 0, -208888, 36835392]\) \(-548347731625/1835008\) \(-3373232128000000\) \([2]\) \(20736\) \(1.8442\)  
2450.t1 2450y6 \([1, 0, 0, -3344888, 2354339392]\) \(2251439055699625/25088\) \(46118408000000\) \([2]\) \(41472\) \(2.1908\)