# Properties

 Label 28224.di Number of curves $4$ Conductor $28224$ CM $$\Q(\sqrt{-3})$$ Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("di1")

sage: E.isogeny_class()

## Elliptic curves in class 28224.di

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
28224.di1 28224di4 $$[0, 0, 0, -26460, -1629936]$$ $$54000$$ $$37940187414528$$ $$[2]$$ $$55296$$ $$1.4000$$   $$-12$$
28224.di2 28224di2 $$[0, 0, 0, -2940, 60368]$$ $$54000$$ $$52044152832$$ $$[2]$$ $$18432$$ $$0.85069$$   $$-12$$
28224.di3 28224di3 $$[0, 0, 0, 0, -74088]$$ $$0$$ $$-2371261713408$$ $$[2]$$ $$27648$$ $$1.0534$$   $$-3$$
28224.di4 28224di1 $$[0, 0, 0, 0, 2744]$$ $$0$$ $$-3252759552$$ $$[2]$$ $$9216$$ $$0.50411$$ $$\Gamma_0(N)$$-optimal $$-3$$

## Rank

sage: E.rank()

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

## Complex multiplication

Each elliptic curve in class 28224.di has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form 28224.2.a.di

sage: E.q_eigenform(10)

$$q + 2 q^{13} - 8 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.