# Properties

 Label 52800ev Number of curves $4$ Conductor $52800$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("ev1")

sage: E.isogeny_class()

## Elliptic curves in class 52800ev

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
52800.dh4 52800ev1 $$[0, -1, 0, 21867, 4565637]$$ $$72268906496/606436875$$ $$-9702990000000000$$ $$$$ $$221184$$ $$1.7490$$ $$\Gamma_0(N)$$-optimal
52800.dh3 52800ev2 $$[0, -1, 0, -315633, 62953137]$$ $$13584145739344/1195803675$$ $$306125740800000000$$ $$$$ $$442368$$ $$2.0955$$
52800.dh2 52800ev3 $$[0, -1, 0, -1562133, 752609637]$$ $$-26348629355659264/24169921875$$ $$-386718750000000000$$ $$$$ $$663552$$ $$2.2983$$
52800.dh1 52800ev4 $$[0, -1, 0, -24999633, 48119797137]$$ $$6749703004355978704/5671875$$ $$1452000000000000$$ $$$$ $$1327104$$ $$2.6448$$

## Rank

sage: E.rank()

The elliptic curves in class 52800ev have rank $$1$$.

## Complex multiplication

The elliptic curves in class 52800ev do not have complex multiplication.

## Modular form 52800.2.a.ev

sage: E.q_eigenform(10)

$$q - q^{3} + 2q^{7} + q^{9} + q^{11} + 2q^{13} + 2q^{19} + O(q^{20})$$ ## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 