# Properties

 Label 4800cd Number of curves 8 Conductor 4800 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("4800.bz1")

sage: E.isogeny_class()

## Elliptic curves in class 4800cd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4800.bz7 4800cd1 [0, 1, 0, -33, -11937] [2] 3072 $$\Gamma_0(N)$$-optimal
4800.bz6 4800cd2 [0, 1, 0, -8033, -275937] [2, 2] 6144
4800.bz4 4800cd3 [0, 1, 0, -128033, -17675937] [2] 12288
4800.bz5 4800cd4 [0, 1, 0, -16033, 356063] [2, 2] 12288
4800.bz2 4800cd5 [0, 1, 0, -216033, 38556063] [2, 2] 24576
4800.bz8 4800cd6 [0, 1, 0, 55967, 2732063] [2] 24576
4800.bz1 4800cd7 [0, 1, 0, -3456033, 2471796063] [2] 49152
4800.bz3 4800cd8 [0, 1, 0, -176033, 53316063] [2] 49152

## Rank

sage: E.rank()

The elliptic curves in class 4800cd have rank $$1$$.

## Modular form4800.2.a.bz

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.