# Properties

 Label 20160bn Number of curves $8$ Conductor $20160$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("20160.bz1")

sage: E.isogeny_class()

## Elliptic curves in class 20160bn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
20160.bz7 20160bn1 [0, 0, 0, -23628, 491888] [2] 73728 $$\Gamma_0(N)$$-optimal
20160.bz5 20160bn2 [0, 0, 0, -207948, -36150928] [2, 2] 147456
20160.bz4 20160bn3 [0, 0, 0, -1544268, 738638192] [2] 221184
20160.bz2 20160bn4 [0, 0, 0, -3318348, -2326649488] [2] 294912
20160.bz6 20160bn5 [0, 0, 0, -46668, -90792592] [2] 294912
20160.bz3 20160bn6 [0, 0, 0, -1555788, 727058288] [2, 2] 442368
20160.bz1 20160bn7 [0, 0, 0, -3715788, -1734477712] [2] 884736
20160.bz8 20160bn8 [0, 0, 0, 419892, 2447480432] [2] 884736

## Rank

sage: E.rank()

The elliptic curves in class 20160bn have rank $$1$$.

## Modular form 20160.2.a.bz

sage: E.q_eigenform(10)

$$q - q^{5} + q^{7} - 2q^{13} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.