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Properties

Label 1050a
Number of curves $8$
Conductor $1050$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("a1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 1050a have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 1050a do not have complex multiplication.

Modular form 1050.2.a.a

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} + q^{9} - q^{12} - 2 q^{13} + q^{14} + q^{16} + 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 1050a

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1050.a7 1050a1 \([1, 1, 0, -1025, -4875]\) \(7633736209/3870720\) \(60480000000\) \([2]\) \(1152\) \(0.76129\) \(\Gamma_0(N)\)-optimal
1050.a5 1050a2 \([1, 1, 0, -9025, 323125]\) \(5203798902289/57153600\) \(893025000000\) \([2, 2]\) \(2304\) \(1.1079\)  
1050.a4 1050a3 \([1, 1, 0, -67025, -6706875]\) \(2131200347946769/2058000\) \(32156250000\) \([2]\) \(3456\) \(1.3106\)  
1050.a2 1050a4 \([1, 1, 0, -144025, 20978125]\) \(21145699168383889/2593080\) \(40516875000\) \([2]\) \(4608\) \(1.4544\)  
1050.a6 1050a5 \([1, 1, 0, -2025, 820125]\) \(-58818484369/18600435000\) \(-290631796875000\) \([2]\) \(4608\) \(1.4544\)  
1050.a3 1050a6 \([1, 1, 0, -67525, -6602375]\) \(2179252305146449/66177562500\) \(1034024414062500\) \([2, 2]\) \(6912\) \(1.6572\)  
1050.a1 1050a7 \([1, 1, 0, -161275, 15616375]\) \(29689921233686449/10380965400750\) \(162202584386718750\) \([2]\) \(13824\) \(2.0037\)  
1050.a8 1050a8 \([1, 1, 0, 18225, -22123125]\) \(42841933504271/13565917968750\) \(-211967468261718750\) \([2]\) \(13824\) \(2.0037\)