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Properties

Label 248430gs
Number of curves $4$
Conductor $248430$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gs1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 248430gs have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 248430gs do not have complex multiplication.

Modular form 248430.2.a.gs

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} + q^{15} + q^{16} + 2 q^{17} + q^{18} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 248430gs

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
248430.gs3 248430gs1 \([1, 1, 1, -1150149278496, -474765911780310207]\) \(296304326013275547793071733369/268420373544960000000\) \(152427676757542258885263360000000\) \([2]\) \(3468165120\) \(5.4736\) \(\Gamma_0(N)\)-optimal
248430.gs2 248430gs2 \([1, 1, 1, -1158459427616, -467557050214169791]\) \(302773487204995438715379645049/8911747415025000000000000\) \(5060707338948561955316025000000000000\) \([2, 2]\) \(6936330240\) \(5.8202\)  
248430.gs1 248430gs3 \([1, 1, 1, -2740596813536, 1085512041941799233]\) \(4008766897254067912673785886329/1423480510711669921875000000\) \(808350812912776687145233154296875000000\) \([2]\) \(13872660480\) \(6.1667\)  
248430.gs4 248430gs4 \([1, 1, 1, 290715572384, -1559258835934169791]\) \(4784981304203817469820354951/1852343836482910078035000000\) \(-1051889108946306554092796393019435000000\) \([2]\) \(13872660480\) \(6.1667\)