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Properties

Label 6762.bl
Number of curves $6$
Conductor $6762$
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, 0, 0, -3877567, 2938562717]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -3877567, 2938562717]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -3877567, 2938562717]) 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 6762.bl have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(7\)\(1\)
\(23\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 6762.bl do not have complex multiplication.

Modular form 6762.2.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);
 
\(q + q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} + q^{8} + q^{9} + 2 q^{10} - 4 q^{11} + q^{12} + 2 q^{13} + 2 q^{15} + q^{16} + 6 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

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 6762.bl

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
6762.bl1 6762bh5 \([1, 0, 0, -3877567, 2938562717]\) \(54804145548726848737/637608031452\) \(75013947292296348\) \([4]\) \(196608\) \(2.3889\)  
6762.bl2 6762bh3 \([1, 0, 0, -867987, -311320143]\) \(614716917569296417/19093020912\) \(2246274817275888\) \([2]\) \(98304\) \(2.0424\)  
6762.bl3 6762bh4 \([1, 0, 0, -248627, 43394385]\) \(14447092394873377/1439452851984\) \(169350188583065616\) \([2, 2]\) \(98304\) \(2.0424\)  
6762.bl4 6762bh2 \([1, 0, 0, -56547, -4433535]\) \(169967019783457/26337394944\) \(3098568177766656\) \([2, 2]\) \(49152\) \(1.6958\)  
6762.bl5 6762bh1 \([1, 0, 0, 6173, -381823]\) \(221115865823/664731648\) \(-78205013655552\) \([2]\) \(24576\) \(1.3492\) \(\Gamma_0(N)\)-optimal
6762.bl6 6762bh6 \([1, 0, 0, 307033, 209981253]\) \(27207619911317663/177609314617308\) \(-20895558255411668892\) \([2]\) \(196608\) \(2.3889\)