# Properties

 Label 100800.gs Number of curves $6$ Conductor $100800$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("100800.gs1")

sage: E.isogeny_class()

## Elliptic curves in class 100800.gs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
100800.gs1 100800dy6 [0, 0, 0, -241920300, 1448293502000] [2] 9437184
100800.gs2 100800dy4 [0, 0, 0, -15120300, 22628702000] [2, 2] 4718592
100800.gs3 100800dy5 [0, 0, 0, -14112300, 25775678000] [2] 9437184
100800.gs4 100800dy3 [0, 0, 0, -5328300, -4474402000] [2] 4718592
100800.gs5 100800dy2 [0, 0, 0, -1008300, 303518000] [2, 2] 2359296
100800.gs6 100800dy1 [0, 0, 0, 143700, 29342000] [2] 1179648 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 100800.gs have rank $$0$$.

## Modular form 100800.2.a.gs

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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.