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Properties

Label 3120.w
Number of curves $6$
Conductor $3120$
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([0, 1, 0, -144240, 21037140]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -144240, 21037140]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -144240, 21037140]) 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 3120.w have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 3120.w do not have complex multiplication.

Modular form 3120.2.a.w

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);
 
\(q + q^{3} + q^{5} + q^{9} - 4 q^{11} + q^{13} + q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

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 3120.w

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
3120.w1 3120z5 \([0, 1, 0, -144240, 21037140]\) \(81025909800741361/11088090\) \(45416816640\) \([4]\) \(12288\) \(1.4572\)  
3120.w2 3120z3 \([0, 1, 0, -13520, -609132]\) \(66730743078481/60937500\) \(249600000000\) \([2]\) \(6144\) \(1.1106\)  
3120.w3 3120z4 \([0, 1, 0, -9040, 324500]\) \(19948814692561/231344100\) \(947585433600\) \([2, 4]\) \(6144\) \(1.1106\)  
3120.w4 3120z6 \([0, 1, 0, -1840, 834260]\) \(-168288035761/73415764890\) \(-300710972989440\) \([4]\) \(12288\) \(1.4572\)  
3120.w5 3120z2 \([0, 1, 0, -1040, -5100]\) \(30400540561/15210000\) \(62300160000\) \([2, 2]\) \(3072\) \(0.76407\)  
3120.w6 3120z1 \([0, 1, 0, 240, -492]\) \(371694959/249600\) \(-1022361600\) \([2]\) \(1536\) \(0.41750\) \(\Gamma_0(N)\)-optimal