# Properties

 Label 100800.pl Number of curves $4$ Conductor $100800$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("pl1")

sage: E.isogeny_class()

## Elliptic curves in class 100800.pl

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
100800.pl1 100800nt4 $$[0, 0, 0, -1008300, -389702000]$$ $$303735479048/105$$ $$39191040000000$$ $$$$ $$1179648$$ $$1.9642$$
100800.pl2 100800nt2 $$[0, 0, 0, -63300, -6032000]$$ $$601211584/11025$$ $$514382400000000$$ $$[2, 2]$$ $$589824$$ $$1.6176$$
100800.pl3 100800nt1 $$[0, 0, 0, -8175, 142000]$$ $$82881856/36015$$ $$26254935000000$$ $$$$ $$294912$$ $$1.2710$$ $$\Gamma_0(N)$$-optimal
100800.pl4 100800nt3 $$[0, 0, 0, -300, -17498000]$$ $$-8/354375$$ $$-132269760000000000$$ $$$$ $$1179648$$ $$1.9642$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 100800.2.a.pl

sage: E.q_eigenform(10)

$$q + q^{7} + 4q^{11} + 6q^{13} - 6q^{17} - 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 