# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 7056.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
7056.z1 7056bg2 $$[0, 0, 0, 0, -756]$$ $$0$$ $$-246903552$$ $$[]$$ $$1728$$ $$0.28926$$   $$-3$$
7056.z2 7056bg1 $$[0, 0, 0, 0, 28]$$ $$0$$ $$-338688$$ $$[]$$ $$576$$ $$-0.26005$$ $$\Gamma_0(N)$$-optimal $$-3$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form7056.2.a.z

sage: E.q_eigenform(10)

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

## 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.