# Properties

 Label 7056.bg Number of curves $4$ Conductor $7056$ CM $$\Q(\sqrt{-7})$$ Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("bg1")

sage: E.isogeny_class()

## Elliptic curves in class 7056.bg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
7056.bg1 7056bq4 $$[0, 0, 0, -262395, 51731946]$$ $$16581375$$ $$120495224844288$$ $$[2]$$ $$28672$$ $$1.7628$$   $$-28$$
7056.bg2 7056bq3 $$[0, 0, 0, -15435, 907578]$$ $$-3375$$ $$-120495224844288$$ $$[2]$$ $$14336$$ $$1.4163$$   $$-7$$
7056.bg3 7056bq2 $$[0, 0, 0, -5355, -150822]$$ $$16581375$$ $$1024192512$$ $$[2]$$ $$4096$$ $$0.78989$$   $$-28$$
7056.bg4 7056bq1 $$[0, 0, 0, -315, -2646]$$ $$-3375$$ $$-1024192512$$ $$[2]$$ $$2048$$ $$0.44332$$ $$\Gamma_0(N)$$-optimal $$-7$$

## Rank

sage: E.rank()

The elliptic curves in class 7056.bg have rank $$0$$.

## Complex multiplication

Each elliptic curve in class 7056.bg has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-7})$$.

## Modular form7056.2.a.bg

sage: E.q_eigenform(10)

$$q + 4 q^{11} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.