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Properties

Label 279414o
Number of curves $4$
Conductor $279414$
CM no
Rank $0$
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, -19495692, 33071508688]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -19495692, 33071508688]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -19495692, 33071508688]) 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 279414o have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(19\)\(1\)
\(43\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 + 11 T^{2}\) 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 279414o do not have complex multiplication.

Modular form 279414.2.a.o

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);
 
\(q - q^{2} + q^{4} - 4 q^{7} - q^{8} - 2 q^{13} + 4 q^{14} + q^{16} + 6 q^{17} + O(q^{20})\) 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 279414o

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
279414.o2 279414o1 \([1, -1, 0, -19495692, 33071508688]\) \(23894093340015625/55042322688\) \(1887756116531427885312\) \([2]\) \(19906560\) \(2.9635\) \(\Gamma_0(N)\)-optimal
279414.o3 279414o2 \([1, -1, 0, -12477852, 57197439040]\) \(-6264610702863625/37578744274608\) \(-1288817420697754042153392\) \([2]\) \(39813120\) \(3.3100\)  
279414.o1 279414o3 \([1, -1, 0, -88943067, -293458381403]\) \(2268876641163765625/228097945239552\) \(7822949146513585836392448\) \([2]\) \(59719680\) \(3.5128\)  
279414.o4 279414o4 \([1, -1, 0, 110675493, -1423179659867]\) \(4371484788393482375/28041364201746432\) \(-961719168135193498761965568\) \([2]\) \(119439360\) \(3.8594\)