# Properties

 Label 27456.z Number of curves $6$ Conductor $27456$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("27456.z1")

sage: E.isogeny_class()

## Elliptic curves in class 27456.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
27456.z1 27456p4 [0, -1, 0, -439297, 112215553] [2] 131072
27456.z2 27456p6 [0, -1, 0, -192577, -31433663] [2] 262144
27456.z3 27456p3 [0, -1, 0, -30337, 1371265] [2, 2] 131072
27456.z4 27456p2 [0, -1, 0, -27457, 1760065] [2, 2] 65536
27456.z5 27456p1 [0, -1, 0, -1537, 33793] [2] 32768 $$\Gamma_0(N)$$-optimal
27456.z6 27456p5 [0, -1, 0, 85823, 9246913] [2] 262144

## Rank

sage: E.rank()

The elliptic curves in class 27456.z have rank $$0$$.

## Modular form 27456.2.a.z

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.