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Properties

Label 350658.bc
Number of curves $4$
Conductor $350658$
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, -49748188617, -4270834644760755]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -49748188617, -4270834644760755]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -49748188617, -4270834644760755]) 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 350658.bc have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(7\)\(1 + T\)
\(11\)\(1\)
\(23\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 350658.bc do not have complex multiplication.

Modular form 350658.2.a.bc

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);
 
\(q - q^{2} + q^{4} - q^{7} - q^{8} + 4 q^{13} + q^{14} + q^{16} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 350658.bc

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
350658.bc1 350658bc4 \([1, -1, 0, -49748188617, -4270834644760755]\) \(10543186518294206197228515625/6611719873695552\) \(8538824436878531065773888\) \([2]\) \(690094080\) \(4.5386\)  
350658.bc2 350658bc3 \([1, -1, 0, -3108670857, -66757841777907]\) \(-2572552807198813678947625/2038409681283182592\) \(-2632540811076729135946395648\) \([2]\) \(345047040\) \(4.1920\)  
350658.bc3 350658bc2 \([1, -1, 0, -626801742, -5604895255536]\) \(21087770069125509765625/1694619018457399188\) \(2188546181995950842348609172\) \([2]\) \(230031360\) \(3.9893\)  
350658.bc4 350658bc1 \([1, -1, 0, 39949398, -396635400540]\) \(5459725204437026375/55780815891710448\) \(-72039137008830216214640112\) \([2]\) \(115015680\) \(3.6427\) \(\Gamma_0(N)\)-optimal