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Properties

Label 338130z
Number of curves $4$
Conductor $338130$
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, -72015, 7538425]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -72015, 7538425]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -72015, 7538425]) 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 338130z have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(13\)\(1 - T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 338130z do not have complex multiplication.

Modular form 338130.2.a.z

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);
 
\(q - q^{2} + q^{4} - q^{5} + 4 q^{7} - q^{8} + q^{10} + q^{13} - 4 q^{14} + q^{16} - 4 q^{19} + O(q^{20})\) 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 338130z

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
338130.z3 338130z1 \([1, -1, 0, -72015, 7538425]\) \(-63378025803/812500\) \(-529517919937500\) \([2]\) \(2654208\) \(1.6345\) \(\Gamma_0(N)\)-optimal
338130.z2 338130z2 \([1, -1, 0, -1155765, 478536175]\) \(261984288445803/42250\) \(27534931836750\) \([2]\) \(5308416\) \(1.9811\)  
338130.z4 338130z3 \([1, -1, 0, 253110, 38194100]\) \(3774555693/3515200\) \(-1670070713708030400\) \([2]\) \(7962624\) \(2.1838\)  
338130.z1 338130z4 \([1, -1, 0, -1307490, 345008060]\) \(520300455507/193072360\) \(91728633950413569720\) \([2]\) \(15925248\) \(2.5304\)