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Properties

Label 6270r
Number of curves $4$
Conductor $6270$
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, -27275, 2000625]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -27275, 2000625]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -27275, 2000625]) 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 6270r have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(11\)\(1 - T\)
\(19\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 6270r do not have complex multiplication.

Modular form 6270.2.a.r

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);
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 2 q^{7} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} - 6 q^{13} - 2 q^{14} + q^{15} + q^{16} - 2 q^{17} + q^{18} - q^{19} + O(q^{20})\) 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.

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

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 6270r

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
6270.r4 6270r1 \([1, 0, 0, -27275, 2000625]\) \(-2243980016705847601/434411683200000\) \(-434411683200000\) \([10]\) \(32000\) \(1.5323\) \(\Gamma_0(N)\)-optimal
6270.r2 6270r2 \([1, 0, 0, -454955, 118072977]\) \(10414276373665867414321/301547812500000\) \(301547812500000\) \([10]\) \(64000\) \(1.8788\)  
6270.r3 6270r3 \([1, 0, 0, -64175, -163709355]\) \(-29229525625065721201/11560253601080069820\) \(-11560253601080069820\) \([2]\) \(160000\) \(2.3370\)  
6270.r1 6270r4 \([1, 0, 0, -4895705, -4128462873]\) \(12976854634417729473922321/148112152782766327650\) \(148112152782766327650\) \([2]\) \(320000\) \(2.6836\)