# Properties

 Label 164730x Number of curves $4$ Conductor $164730$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("164730.cc1")

sage: E.isogeny_class()

## Elliptic curves in class 164730x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
164730.cc4 164730x1 [1, 1, 1, -2896, 101153] [2] 491520 $$\Gamma_0(N)$$-optimal
164730.cc3 164730x2 [1, 1, 1, -54916, 4928609] [2, 2] 983040
164730.cc1 164730x3 [1, 1, 1, -878566, 316597769] [2] 1966080
164730.cc2 164730x4 [1, 1, 1, -63586, 3257033] [2] 1966080

## Rank

sage: E.rank()

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

## Modular form 164730.2.a.cc

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} + q^{8} + q^{9} - q^{10} + 4q^{11} - q^{12} - 2q^{13} - 4q^{14} + q^{15} + q^{16} + q^{18} - q^{19} + O(q^{20})$$

## 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.