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Properties

Label 279174h
Number of curves $4$
Conductor $279174$
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, 1342255, 4215948693]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, 1342255, 4215948693]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, 1342255, 4215948693]) 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 279174h have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(7\)\(1 + T\)
\(17\)\(1\)
\(23\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 279174h do not have complex multiplication.

Modular form 279174.2.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);
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} + q^{9} - 6 q^{11} - q^{12} + 2 q^{13} + q^{14} + q^{16} - q^{18} + 2 q^{19} + O(q^{20})\) 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.

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

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 279174h

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
279174.h4 279174h1 \([1, 1, 0, 1342255, 4215948693]\) \(11079872671250375/324440155855872\) \(-7831196648341864455168\) \([2]\) \(22118400\) \(2.8811\) \(\Gamma_0(N)\)-optimal
279174.h2 279174h2 \([1, 1, 0, -32366705, 67514633781]\) \(155355156733986861625/8291568305839392\) \(200138302100411427318048\) \([2]\) \(44236800\) \(3.2277\)  
279174.h3 279174h3 \([1, 1, 0, -12117920, -115820276736]\) \(-8152944444844179625/235342826399858688\) \(-5680603710881610671849472\) \([2]\) \(66355200\) \(3.4304\)  
279174.h1 279174h4 \([1, 1, 0, -438265760, -3514519759872]\) \(385693937170561837203625/2159357734550274048\) \(52121646313390923802509312\) \([2]\) \(132710400\) \(3.7770\)