# Properties

 Label 162450dq Number of curves $4$ Conductor $162450$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("dq1")

sage: E.isogeny_class()

## Elliptic curves in class 162450dq

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162450.w3 162450dq1 $$[1, -1, 0, -2519667, 1530856741]$$ $$3301293169/22800$$ $$12218109332456250000$$ $$[2]$$ $$4423680$$ $$2.4960$$ $$\Gamma_0(N)$$-optimal
162450.w2 162450dq2 $$[1, -1, 0, -4144167, -683336759]$$ $$14688124849/8122500$$ $$4352701449687539062500$$ $$[2, 2]$$ $$8847360$$ $$2.8426$$
162450.w4 162450dq3 $$[1, -1, 0, 16162083, -5414693009]$$ $$871257511151/527800050$$ $$-282838540200696288281250$$ $$[2]$$ $$17694720$$ $$3.1892$$
162450.w1 162450dq4 $$[1, -1, 0, -50442417, -137679858509]$$ $$26487576322129/44531250$$ $$23863494789953613281250$$ $$[2]$$ $$17694720$$ $$3.1892$$

## Rank

sage: E.rank()

The elliptic curves in class 162450dq have rank $$1$$.

## Complex multiplication

The elliptic curves in class 162450dq do not have complex multiplication.

## Modular form 162450.2.a.dq

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{8} - 4q^{11} + 2q^{13} + q^{16} + 2q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.