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Properties

Label 4950be
Number of curves $6$
Conductor $4950$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("be1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 4950be have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1\)
\(11\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + T + 7 T^{2}\) 1.7.b
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 + 7 T + 19 T^{2}\) 1.19.h
\(23\) \( 1 + 3 T + 23 T^{2}\) 1.23.d
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 4950be do not have complex multiplication.

Modular form 4950.2.a.be

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{8} - q^{11} - 6 q^{13} + q^{16} + 2 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 4950be

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4950.bg6 4950be1 \([1, -1, 1, 57370, -517003]\) \(1833318007919/1070530560\) \(-12194012160000000\) \([4]\) \(36864\) \(1.7761\) \(\Gamma_0(N)\)-optimal
4950.bg5 4950be2 \([1, -1, 1, -230630, -3973003]\) \(119102750067601/68309049600\) \(778082768100000000\) \([2, 2]\) \(73728\) \(2.1226\)  
4950.bg2 4950be3 \([1, -1, 1, -2660630, -1666093003]\) \(182864522286982801/463015182960\) \(5274032318403750000\) \([2]\) \(147456\) \(2.4692\)  
4950.bg3 4950be4 \([1, -1, 1, -2408630, 1433506997]\) \(135670761487282321/643043610000\) \(7324668620156250000\) \([2, 2]\) \(147456\) \(2.4692\)  
4950.bg1 4950be5 \([1, -1, 1, -38494130, 91935940997]\) \(553808571467029327441/12529687500\) \(142720971679687500\) \([2]\) \(294912\) \(2.8158\)  
4950.bg4 4950be6 \([1, -1, 1, -1171130, 2903656997]\) \(-15595206456730321/310672490129100\) \(-3538753832876779687500\) \([2]\) \(294912\) \(2.8158\)