# Properties

 Label 100800mg Number of curves $4$ Conductor $100800$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("mg1")

sage: E.isogeny_class()

## Elliptic curves in class 100800mg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
100800.p4 100800mg1 $$[0, 0, 0, 33300, -3726000]$$ $$1367631/2800$$ $$-8360755200000000$$ $$$$ $$589824$$ $$1.7396$$ $$\Gamma_0(N)$$-optimal
100800.p3 100800mg2 $$[0, 0, 0, -254700, -40014000]$$ $$611960049/122500$$ $$365783040000000000$$ $$[2, 2]$$ $$1179648$$ $$2.0862$$
100800.p2 100800mg3 $$[0, 0, 0, -1262700, 510354000]$$ $$74565301329/5468750$$ $$16329600000000000000$$ $$$$ $$2359296$$ $$2.4327$$
100800.p1 100800mg4 $$[0, 0, 0, -3854700, -2912814000]$$ $$2121328796049/120050$$ $$358467379200000000$$ $$$$ $$2359296$$ $$2.4327$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 100800mg do not have complex multiplication.

## Modular form 100800.2.a.mg

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. 