# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("3120.w1")

sage: E.isogeny_class()

## Elliptic curves in class 3120.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3120.w1 3120z5 [0, 1, 0, -144240, 21037140] [4] 12288
3120.w2 3120z3 [0, 1, 0, -13520, -609132] [2] 6144
3120.w3 3120z4 [0, 1, 0, -9040, 324500] [2, 4] 6144
3120.w4 3120z6 [0, 1, 0, -1840, 834260] [4] 12288
3120.w5 3120z2 [0, 1, 0, -1040, -5100] [2, 2] 3072
3120.w6 3120z1 [0, 1, 0, 240, -492] [2] 1536 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 3120.w have rank $$1$$.

## Modular form3120.2.a.w

sage: E.q_eigenform(10)

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

## 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.