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Properties

Label 27378b
Number of curves $4$
Conductor $27378$
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, -1, 0, -792, 9192]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -792, 9192]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -792, 9192]) 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 27378b have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 3 T + 5 T^{2}\) 1.5.d
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 - T + 19 T^{2}\) 1.19.ab
\(23\) \( 1 + 9 T + 23 T^{2}\) 1.23.j
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 27378b do not have complex multiplication.

Modular form 27378.2.a.b

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);
 
\(q - q^{2} + q^{4} - 2 q^{7} - q^{8} + 3 q^{11} + 2 q^{14} + q^{16} - 3 q^{17} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 7 & 21 \\ 3 & 1 & 21 & 7 \\ 7 & 21 & 1 & 3 \\ 21 & 7 & 3 & 1 \end{array}\right)\)

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 27378b

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
27378.f3 27378b1 \([1, -1, 0, -792, 9192]\) \(-140625/8\) \(-3127772232\) \([]\) \(13824\) \(0.57889\) \(\Gamma_0(N)\)-optimal
27378.f4 27378b2 \([1, -1, 0, 4278, 15614]\) \(3375/2\) \(-5130328403538\) \([]\) \(41472\) \(1.1282\)  
27378.f2 27378b3 \([1, -1, 0, -16002, -1578732]\) \(-1159088625/2097152\) \(-819926723985408\) \([]\) \(96768\) \(1.5518\)  
27378.f1 27378b4 \([1, -1, 0, -1638402, -806786668]\) \(-189613868625/128\) \(-328341017826432\) \([]\) \(290304\) \(2.1012\)