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Properties

Label 68450bl
Number of curves $4$
Conductor $68450$
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, 0, 0, -713, -86533]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -713, -86533]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -713, -86533]) 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 68450bl have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(5\)\(1\)
\(37\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 3 T^{2}\) 1.3.a
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 3 T + 19 T^{2}\) 1.19.ad
\(23\) \( 1 + T + 23 T^{2}\) 1.23.b
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 68450bl do not have complex multiplication.

Modular form 68450.2.a.bl

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);
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} + 2 q^{7} + q^{8} - 2 q^{9} - 3 q^{11} + q^{12} + 4 q^{13} + 2 q^{14} + q^{16} + 3 q^{17} - 2 q^{18} - 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 5 & 15 \\ 3 & 1 & 15 & 5 \\ 5 & 15 & 1 & 3 \\ 15 & 5 & 3 & 1 \end{array}\right)\)

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 68450bl

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
68450.bm3 68450bl1 \([1, 0, 0, -713, -86533]\) \(-25/2\) \(-3207158011250\) \([]\) \(103680\) \(1.0794\) \(\Gamma_0(N)\)-optimal
68450.bm1 68450bl2 \([1, 0, 0, -171838, -27432308]\) \(-349938025/8\) \(-12828632045000\) \([]\) \(311040\) \(1.6287\)  
68450.bm2 68450bl3 \([1, 0, 0, -103388, 15417392]\) \(-121945/32\) \(-32071580112500000\) \([]\) \(518400\) \(1.8841\)  
68450.bm4 68450bl4 \([1, 0, 0, 752237, -113781983]\) \(46969655/32768\) \(-32841298035200000000\) \([]\) \(1555200\) \(2.4334\)