# Properties

 Label 38640.r Number of curves $4$ Conductor $38640$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("r1")

sage: E.isogeny_class()

## Elliptic curves in class 38640.r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.r1 38640bo4 $$[0, -1, 0, -29496, -1810704]$$ $$692895692874169/51420783750$$ $$210619530240000$$ $$$$ $$147456$$ $$1.4942$$
38640.r2 38640bo2 $$[0, -1, 0, -5976, 146160]$$ $$5763259856089/1143116100$$ $$4682203545600$$ $$[2, 2]$$ $$73728$$ $$1.1476$$
38640.r3 38640bo1 $$[0, -1, 0, -5656, 165616]$$ $$4886171981209/270480$$ $$1107886080$$ $$$$ $$36864$$ $$0.80100$$ $$\Gamma_0(N)$$-optimal
38640.r4 38640bo3 $$[0, -1, 0, 12424, 852720]$$ $$51774168853511/107398242630$$ $$-439903201812480$$ $$$$ $$147456$$ $$1.4942$$

## Rank

sage: E.rank()

The elliptic curves in class 38640.r have rank $$1$$.

## Complex multiplication

The elliptic curves in class 38640.r do not have complex multiplication.

## Modular form 38640.2.a.r

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{7} + q^{9} - 4 q^{11} + 2 q^{13} + q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})$$ ## 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. 