# Properties

 Label 31824bk Number of curves $6$ Conductor $31824$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("31824.bg1")

sage: E.isogeny_class()

## Elliptic curves in class 31824bk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
31824.bg4 31824bk1 [0, 0, 0, -77619, 8323378] [2] 65536 $$\Gamma_0(N)$$-optimal
31824.bg3 31824bk2 [0, 0, 0, -78339, 8161090] [2, 2] 131072
31824.bg5 31824bk3 [0, 0, 0, 31821, 29289778] [2] 262144
31824.bg2 31824bk4 [0, 0, 0, -200019, -23354030] [2, 2] 262144
31824.bg6 31824bk5 [0, 0, 0, 558141, -158154878] [2] 524288
31824.bg1 31824bk6 [0, 0, 0, -2905059, -1905520862] [2] 524288

## Rank

sage: E.rank()

The elliptic curves in class 31824bk have rank $$0$$.

## Modular form 31824.2.a.bg

sage: E.q_eigenform(10)

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

## 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.