# Properties

 Label 5292.f Number of curves $2$ Conductor $5292$ CM $$\Q(\sqrt{-3})$$ Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 5292.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
5292.f1 5292b1 $$[0, 0, 0, 0, -67228]$$ $$0$$ $$-1952468921088$$ $$[]$$ $$4536$$ $$1.0372$$ $$\Gamma_0(N)$$-optimal $$-3$$
5292.f2 5292b2 $$[0, 0, 0, 0, 1815156]$$ $$0$$ $$-1423349843473152$$ $$[]$$ $$13608$$ $$1.5865$$   $$-3$$

## Rank

sage: E.rank()

The elliptic curves in class 5292.f have rank $$1$$.

## Complex multiplication

Each elliptic curve in class 5292.f has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form5292.2.a.f

sage: E.q_eigenform(10)

$$q - 2q^{13} + q^{19} + O(q^{20})$$ ## 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. 