# Properties

 Label 39984bv Number of curves $6$ Conductor $39984$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("39984.bj1")

sage: E.isogeny_class()

## Elliptic curves in class 39984bv

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
39984.bj5 39984bv1 [0, -1, 0, -55071312, 157124288448] [2] 4423680 $$\Gamma_0(N)$$-optimal
39984.bj4 39984bv2 [0, -1, 0, -71127632, 58050371520] [2, 2] 8847360
39984.bj6 39984bv3 [0, -1, 0, 274334128, 456298688448] [2] 17694720
39984.bj2 39984bv4 [0, -1, 0, -673490512, -6681185529920] [2, 2] 17694720
39984.bj3 39984bv5 [0, -1, 0, -229401552, -15360815435328] [2] 35389440
39984.bj1 39984bv6 [0, -1, 0, -10755385552, -429322291122752] [2] 35389440

## Rank

sage: E.rank()

The elliptic curves in class 39984bv have rank $$1$$.

## Modular form 39984.2.a.bj

sage: E.q_eigenform(10)

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

## 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.