# Properties

 Label 216384l Number of curves $6$ Conductor $216384$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("216384.fi1")

sage: E.isogeny_class()

## Elliptic curves in class 216384l

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
216384.fi5 216384l1 [0, 1, 0, 395071, 195888447] [2] 4718592 $$\Gamma_0(N)$$-optimal
216384.fi4 216384l2 [0, 1, 0, -3619009, 2266350911] [2, 2] 9437184
216384.fi2 216384l3 [0, 1, 0, -55551169, 159340362047] [2] 18874368
216384.fi3 216384l4 [0, 1, 0, -15912129, -22233837249] [2, 2] 18874368
216384.fi6 216384l5 [0, 1, 0, 19650111, -107490751425] [2] 37748736
216384.fi1 216384l6 [0, 1, 0, -248164289, -1504792275393] [2] 37748736

## Rank

sage: E.rank()

The elliptic curves in class 216384l have rank $$1$$.

## Modular form 216384.2.a.fi

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.