# Properties

 Label 81120u Number of curves $4$ Conductor $81120$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("u1")

sage: E.isogeny_class()

## Elliptic curves in class 81120u

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
81120.bw3 81120u1 $$[0, 1, 0, -11210, -429792]$$ $$504358336/38025$$ $$11746522382400$$ $$[2, 2]$$ $$129024$$ $$1.2531$$ $$\Gamma_0(N)$$-optimal
81120.bw4 81120u2 $$[0, 1, 0, 10760, -1888600]$$ $$55742968/658125$$ $$-1626441560640000$$ $$[4]$$ $$258048$$ $$1.5997$$
81120.bw2 81120u3 $$[0, 1, 0, -36560, 2176188]$$ $$2186875592/428415$$ $$1058753217400320$$ $$[2]$$ $$258048$$ $$1.5997$$
81120.bw1 81120u4 $$[0, 1, 0, -175985, -28474497]$$ $$30488290624/195$$ $$3855268884480$$ $$[2]$$ $$258048$$ $$1.5997$$

## Rank

sage: E.rank()

The elliptic curves in class 81120u have rank $$1$$.

## Complex multiplication

The elliptic curves in class 81120u do not have complex multiplication.

## Modular form 81120.2.a.u

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} + q^{9} + q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.