# Properties

 Label 7056.bf Number of curves $2$ Conductor $7056$ CM $$\Q(\sqrt{-3})$$ Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bf1")

sage: E.isogeny_class()

## Elliptic curves in class 7056.bf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
7056.bf1 7056bh2 $$[0, 0, 0, 0, -7260624]$$ $$0$$ $$-22773597495570432$$ $$[]$$ $$36288$$ $$1.8176$$   $$-3$$
7056.bf2 7056bh1 $$[0, 0, 0, 0, 268912]$$ $$0$$ $$-31239502737408$$ $$[]$$ $$12096$$ $$1.2683$$ $$\Gamma_0(N)$$-optimal $$-3$$

## Rank

sage: E.rank()

The elliptic curves in class 7056.bf have rank $$1$$.

## Complex multiplication

Each elliptic curve in class 7056.bf has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form7056.2.a.bf

sage: E.q_eigenform(10)

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