# Properties

 Label 164730d Number of curves $4$ Conductor $164730$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("164730.dj1")

sage: E.isogeny_class()

## Elliptic curves in class 164730d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
164730.dj3 164730d1 [1, 0, 0, -8965, 324017]  327680 $$\Gamma_0(N)$$-optimal
164730.dj2 164730d2 [1, 0, 0, -14745, -146475] [2, 2] 655360
164730.dj4 164730d3 [1, 0, 0, 57505, -1143525]  1310720
164730.dj1 164730d4 [1, 0, 0, -179475, -29237793]  1310720

## Rank

sage: E.rank()

The elliptic curves in class 164730d have rank $$0$$.

## Modular form 164730.2.a.dj

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} - 4q^{11} + q^{12} + 2q^{13} + q^{15} + q^{16} + q^{18} - q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 