# Properties

 Label 369600qd Number of curves $4$ Conductor $369600$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("qd1")

sage: E.isogeny_class()

## Elliptic curves in class 369600qd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
369600.qd3 369600qd1 $$[0, 1, 0, -266133, 53538363]$$ $$-130287139815424/2250652635$$ $$-36010442160000000$$ $$[2]$$ $$3981312$$ $$1.9748$$ $$\Gamma_0(N)$$-optimal
369600.qd2 369600qd2 $$[0, 1, 0, -4275633, 3401470863]$$ $$33766427105425744/9823275$$ $$2514758400000000$$ $$[2]$$ $$7962624$$ $$2.3214$$
369600.qd4 369600qd3 $$[0, 1, 0, 1029867, 257334363]$$ $$7549996227362816/6152409907875$$ $$-98438558526000000000$$ $$[2]$$ $$11943936$$ $$2.5241$$
369600.qd1 369600qd4 $$[0, 1, 0, -4959633, 2239858863]$$ $$52702650535889104/22020583921875$$ $$5637269484000000000000$$ $$[2]$$ $$23887872$$ $$2.8707$$

## Rank

sage: E.rank()

The elliptic curves in class 369600qd have rank $$1$$.

## Complex multiplication

The elliptic curves in class 369600qd do not have complex multiplication.

## Modular form 369600.2.a.qd

sage: E.q_eigenform(10)

$$q + q^{3} - q^{7} + q^{9} + q^{11} + 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.