# Properties

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

# Related objects

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 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.