# Properties

 Label 2142d Number of curves $4$ Conductor $2142$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 2142d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2142.g4 2142d1 $$[1, -1, 0, 9, -2723]$$ $$103823/4386816$$ $$-3197988864$$ $$[2]$$ $$1536$$ $$0.50271$$ $$\Gamma_0(N)$$-optimal
2142.g3 2142d2 $$[1, -1, 0, -2871, -57443]$$ $$3590714269297/73410624$$ $$53516344896$$ $$[2, 2]$$ $$3072$$ $$0.84928$$
2142.g1 2142d3 $$[1, -1, 0, -45711, -3750251]$$ $$14489843500598257/6246072$$ $$4553386488$$ $$[2]$$ $$6144$$ $$1.1959$$
2142.g2 2142d4 $$[1, -1, 0, -6111, 98725]$$ $$34623662831857/14438442312$$ $$10525624445448$$ $$[2]$$ $$6144$$ $$1.1959$$

## Rank

sage: E.rank()

The elliptic curves in class 2142d have rank $$0$$.

## Complex multiplication

The elliptic curves in class 2142d do not have complex multiplication.

## Modular form2142.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + 2q^{5} - q^{7} - q^{8} - 2q^{10} - 6q^{13} + q^{14} + q^{16} - q^{17} + O(q^{20})$$

## 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.