# Properties

 Label 4851o Number of curves 6 Conductor 4851 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("4851.p1")

sage: E.isogeny_class()

## Elliptic curves in class 4851o

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4851.p4 4851o1 [1, -1, 0, -15003, 710856] [2] 7680 $$\Gamma_0(N)$$-optimal
4851.p3 4851o2 [1, -1, 0, -17208, 489915] [2, 2] 15360
4851.p2 4851o3 [1, -1, 0, -125253, -16689240] [2, 2] 30720
4851.p6 4851o4 [1, -1, 0, 55557, 3502386] [2] 30720
4851.p1 4851o5 [1, -1, 0, -1992888, -1082361771] [2] 61440
4851.p5 4851o6 [1, -1, 0, 13662, -51779169] [2] 61440

## Rank

sage: E.rank()

The elliptic curves in class 4851o have rank $$0$$.

## Modular form4851.2.a.p

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} + q^{11} - 6q^{13} - q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.