# Properties

 Label 28224.fr Number of curves $2$ Conductor $28224$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("fr1")

sage: E.isogeny_class()

## Elliptic curves in class 28224.fr

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.fr1 28224e1 $$[0, 0, 0, -292236, 60812528]$$ $$-67645179/8$$ $$-326420926562304$$ $$[]$$ $$193536$$ $$1.8105$$ $$\Gamma_0(N)$$-optimal
28224.fr2 28224e2 $$[0, 0, 0, 37044, 187738992]$$ $$189/512$$ $$-15229494749690855424$$ $$[]$$ $$580608$$ $$2.3598$$

## Rank

sage: E.rank()

The elliptic curves in class 28224.fr have rank $$1$$.

## Complex multiplication

The elliptic curves in class 28224.fr do not have complex multiplication.

## Modular form 28224.2.a.fr

sage: E.q_eigenform(10)

$$q + 3q^{5} - 3q^{11} - 2q^{13} - 6q^{17} - 2q^{19} + O(q^{20})$$ ## 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. 