# Properties

 Label 1568.e Number of curves $4$ Conductor $1568$ CM $$\Q(\sqrt{-1})$$ Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 1568.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
1568.e1 1568g2 $$[0, 0, 0, -539, -4802]$$ $$287496$$ $$60236288$$ $$[2]$$ $$384$$ $$0.35557$$   $$-16$$
1568.e2 1568g3 $$[0, 0, 0, -539, 4802]$$ $$287496$$ $$60236288$$ $$[2]$$ $$384$$ $$0.35557$$   $$-16$$
1568.e3 1568g1 $$[0, 0, 0, -49, 0]$$ $$1728$$ $$7529536$$ $$[2, 2]$$ $$192$$ $$0.0089957$$ $$\Gamma_0(N)$$-optimal $$-4$$
1568.e4 1568g4 $$[0, 0, 0, 196, 0]$$ $$1728$$ $$-481890304$$ $$[2]$$ $$384$$ $$0.35557$$   $$-4$$

## Rank

sage: E.rank()

The elliptic curves in class 1568.e have rank $$1$$.

## Complex multiplication

Each elliptic curve in class 1568.e has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-1})$$.

## Modular form1568.2.a.e

sage: E.q_eigenform(10)

$$q + 2q^{5} - 3q^{9} - 6q^{13} - 2q^{17} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.