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Properties

Label 153450bk
Number of curves $4$
Conductor $153450$
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, 1, -109501880, -440925869253]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -109501880, -440925869253]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -109501880, -440925869253]) 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 153450bk have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1\)
\(11\)\(1 - T\)
\(31\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 - 4 T + 29 T^{2}\) 1.29.ae
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 153450bk do not have complex multiplication.

Modular form 153450.2.a.bk

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);
 
\(q + q^{2} + q^{4} + 2 q^{7} + q^{8} - q^{11} + 6 q^{13} + 2 q^{14} + q^{16} + 8 q^{17} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

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 153450bk

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
153450.ds3 153450bk1 \([1, -1, 1, -109501880, -440925869253]\) \(12747965531857798561201/2986780262400000\) \(34021293926400000000000\) \([2]\) \(33792000\) \(3.3139\) \(\Gamma_0(N)\)-optimal
153450.ds4 153450bk2 \([1, -1, 1, -96829880, -546863789253]\) \(-8814635019030000319921/6242069790000000000\) \(-71101076201718750000000000\) \([2]\) \(67584000\) \(3.6605\)  
153450.ds1 153450bk3 \([1, -1, 1, -1964671880, 33448711650747]\) \(73628549562506871957390001/178215946908754500240\) \(2029991020257531729296250000\) \([2]\) \(168960000\) \(4.1186\)  
153450.ds2 153450bk4 \([1, -1, 1, -1239942380, 58437384810747]\) \(-18508902577171306222471921/118801759721890483665900\) \(-1353226294332158790506892187500\) \([2]\) \(337920000\) \(4.4652\)