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Properties

Label 240198h
Number of curves $4$
Conductor $240198$
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, 1, 0, -294025, -61365947]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -294025, -61365947]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -294025, -61365947]) 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 240198h have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(7\)\(1\)
\(19\)\(1 + T\)
\(43\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 240198h do not have complex multiplication.

Modular form 240198.2.a.h

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

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 240198h

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
240198.h2 240198h1 \([1, 1, 0, -294025, -61365947]\) \(23894093340015625/55042322688\) \(6475674221920512\) \([2]\) \(1990656\) \(1.9149\) \(\Gamma_0(N)\)-optimal
240198.h3 240198h2 \([1, 1, 0, -188185, -106009259]\) \(-6264610702863625/37578744274608\) \(-4421101685163356592\) \([2]\) \(3981312\) \(2.2615\)  
240198.h1 240198h3 \([1, 1, 0, -1341400, 543002944]\) \(2268876641163765625/228097945239552\) \(26835495159488053248\) \([2]\) \(5971968\) \(2.4642\)  
240198.h4 240198h4 \([1, 1, 0, 1669160, 2636546368]\) \(4371484788393482375/28041364201746432\) \(-3299038456971265978368\) \([2]\) \(11943936\) \(2.8108\)