Properties

Label 3120.w
Number of curves $6$
Conductor $3120$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3120.w

sage: E.isogeny_class().curves
 
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

Rank

sage: E.rank()
 

The elliptic curves in class 3120.w have rank \(1\).

Complex multiplication

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

Modular form 3120.2.a.w

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{9} - 4q^{11} + q^{13} + q^{15} - 6q^{17} - 4q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) 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

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.