# Properties

 Label 2299d Number of curves $3$ Conductor $2299$ CM no Rank $0$ Graph

# Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 2299d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2299.b3 2299d1 $$[0, 1, 1, 81, 39]$$ $$32768/19$$ $$-33659659$$ $$[]$$ $$450$$ $$0.13377$$ $$\Gamma_0(N)$$-optimal
2299.b2 2299d2 $$[0, 1, 1, -1129, 15164]$$ $$-89915392/6859$$ $$-12151136899$$ $$[]$$ $$1350$$ $$0.68308$$
2299.b1 2299d3 $$[0, 1, 1, -93089, 10900929]$$ $$-50357871050752/19$$ $$-33659659$$ $$[]$$ $$4050$$ $$1.2324$$

## Rank

sage: E.rank()

The elliptic curves in class 2299d have rank $$0$$.

## Complex multiplication

The elliptic curves in class 2299d do not have complex multiplication.

## Modular form2299.2.a.d

sage: E.q_eigenform(10)

$$q - 2q^{3} - 2q^{4} + 3q^{5} + q^{7} + q^{9} + 4q^{12} + 4q^{13} - 6q^{15} + 4q^{16} + 3q^{17} - q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.