# Properties

 Label 100800mg Number of curves 4 Conductor 100800 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("100800.p1")

sage: E.isogeny_class()

## Elliptic curves in class 100800mg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
100800.p4 100800mg1 [0, 0, 0, 33300, -3726000]  589824 $$\Gamma_0(N)$$-optimal
100800.p3 100800mg2 [0, 0, 0, -254700, -40014000] [2, 2] 1179648
100800.p2 100800mg3 [0, 0, 0, -1262700, 510354000]  2359296
100800.p1 100800mg4 [0, 0, 0, -3854700, -2912814000]  2359296

## Rank

sage: E.rank()

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

## Modular form 100800.2.a.p

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 