# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("100800.gr1")

sage: E.isogeny_class()

## Elliptic curves in class 100800.gr

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
100800.gr1 100800lw6 [0, 0, 0, -9432300, -11149742000] [2] 3145728
100800.gr2 100800lw4 [0, 0, 0, -612300, -160022000] [2, 2] 1572864
100800.gr3 100800lw2 [0, 0, 0, -162300, 22678000] [2, 2] 786432
100800.gr4 100800lw1 [0, 0, 0, -157800, 24127000] [2] 393216 $$\Gamma_0(N)$$-optimal
100800.gr5 100800lw3 [0, 0, 0, 215700, 112642000] [2] 1572864
100800.gr6 100800lw5 [0, 0, 0, 1007700, -863102000] [2] 3145728

## Rank

sage: E.rank()

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

## Modular form 100800.2.a.gr

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.