# Properties

 Label 162450.w Number of curves $4$ Conductor $162450$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("162450.w1")

sage: E.isogeny_class()

## Elliptic curves in class 162450.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
162450.w1 162450dq4 [1, -1, 0, -50442417, -137679858509]  17694720
162450.w2 162450dq2 [1, -1, 0, -4144167, -683336759] [2, 2] 8847360
162450.w3 162450dq1 [1, -1, 0, -2519667, 1530856741]  4423680 $$\Gamma_0(N)$$-optimal
162450.w4 162450dq3 [1, -1, 0, 16162083, -5414693009]  17694720

## Rank

sage: E.rank()

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

## Modular form 162450.2.a.w

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 LMFDB 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 LMFDB labels. 