# Properties

 Label 27456.cf Number of curves $6$ Conductor $27456$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("27456.cf1")

sage: E.isogeny_class()

## Elliptic curves in class 27456.cf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
27456.cf1 27456bz4 [0, 1, 0, -439297, -112215553] [2] 131072
27456.cf2 27456bz6 [0, 1, 0, -192577, 31433663] [2] 262144
27456.cf3 27456bz3 [0, 1, 0, -30337, -1371265] [2, 2] 131072
27456.cf4 27456bz2 [0, 1, 0, -27457, -1760065] [2, 2] 65536
27456.cf5 27456bz1 [0, 1, 0, -1537, -33793] [2] 32768 $$\Gamma_0(N)$$-optimal
27456.cf6 27456bz5 [0, 1, 0, 85823, -9246913] [2] 262144

## Rank

sage: E.rank()

The elliptic curves in class 27456.cf have rank $$1$$.

## Modular form 27456.2.a.cf

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.