# Properties

 Label 10830g Number of curves $4$ Conductor $10830$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

## Elliptic curves in class 10830g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10830.g4 10830g1 $$[1, 1, 0, 3369928, 5064449856]$$ $$89962967236397039/287450726400000$$ $$-13523372667577958400000$$ $$$$ $$864000$$ $$2.9298$$ $$\Gamma_0(N)$$-optimal
10830.g3 10830g2 $$[1, 1, 0, -31748152, 59307836224]$$ $$75224183150104868881/11219310000000000$$ $$527822323162110000000000$$ $$$$ $$1728000$$ $$3.2763$$
10830.g2 10830g3 $$[1, 1, 0, -1191828872, 15836364428016]$$ $$-3979640234041473454886161/1471455901872240$$ $$-69225939256229080243440$$ $$$$ $$4320000$$ $$3.7345$$
10830.g1 10830g4 $$[1, 1, 0, -19069263652, 1013550971091124]$$ $$16300610738133468173382620881/2228489100$$ $$104841233008397100$$ $$$$ $$8640000$$ $$4.0810$$

## Rank

sage: E.rank()

The elliptic curves in class 10830g have rank $$1$$.

## Complex multiplication

The elliptic curves in class 10830g do not have complex multiplication.

## Modular form 10830.2.a.g

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} - q^{10} + 2q^{11} - q^{12} - 4q^{13} + 2q^{14} - q^{15} + q^{16} - 2q^{17} - q^{18} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 