# Properties

 Label 100800lz Number of curves $6$ Conductor $100800$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("100800.hj1")

sage: E.isogeny_class()

## Elliptic curves in class 100800lz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
100800.hj5 100800lz1 [0, 0, 0, -57900, 11842000] [2] 786432 $$\Gamma_0(N)$$-optimal
100800.hj4 100800lz2 [0, 0, 0, -1209900, 511810000] [2, 2] 1572864
100800.hj3 100800lz3 [0, 0, 0, -1497900, 249730000] [2, 2] 3145728
100800.hj1 100800lz4 [0, 0, 0, -19353900, 32771842000] [2] 3145728
100800.hj6 100800lz5 [0, 0, 0, 5558100, 1929058000] [2] 6291456
100800.hj2 100800lz6 [0, 0, 0, -13161900, -18202718000] [2] 6291456

## Rank

sage: E.rank()

The elliptic curves in class 100800lz have rank $$1$$.

## Modular form 100800.2.a.hj

sage: E.q_eigenform(10)

$$q - q^{7} + 4q^{11} + 6q^{13} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.