# Properties

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

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 164730.dj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
164730.dj1 164730d4 $$[1, 0, 0, -179475, -29237793]$$ $$26487576322129/44531250$$ $$1074876119531250$$ $$[2]$$ $$1310720$$ $$1.7795$$
164730.dj2 164730d2 $$[1, 0, 0, -14745, -146475]$$ $$14688124849/8122500$$ $$196057404202500$$ $$[2, 2]$$ $$655360$$ $$1.4330$$
164730.dj3 164730d1 $$[1, 0, 0, -8965, 324017]$$ $$3301293169/22800$$ $$550336573200$$ $$[2]$$ $$327680$$ $$1.0864$$ $$\Gamma_0(N)$$-optimal
164730.dj4 164730d3 $$[1, 0, 0, 57505, -1143525]$$ $$871257511151/527800050$$ $$-12739810125078450$$ $$[2]$$ $$1310720$$ $$1.7795$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 164730.dj do not have complex multiplication.

## Modular form 164730.2.a.dj

sage: E.q_eigenform(10)

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

## 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 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.