# Properties

 Label 2700.b Number of curves $2$ Conductor $2700$ CM $$\Q(\sqrt{-3})$$ Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 2700.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
2700.b1 2700p2 $$[0, 0, 0, 0, -13500]$$ $$0$$ $$-78732000000$$ $$[]$$ $$2592$$ $$0.76966$$   $$-3$$
2700.b2 2700p1 $$[0, 0, 0, 0, 500]$$ $$0$$ $$-108000000$$ $$[]$$ $$864$$ $$0.22035$$ $$\Gamma_0(N)$$-optimal $$-3$$

## Rank

sage: E.rank()

The elliptic curves in class 2700.b have rank $$1$$.

## Complex multiplication

Each elliptic curve in class 2700.b has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form2700.2.a.b

sage: E.q_eigenform(10)

$$q - 5 q^{7} + 7 q^{13} - q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.