# Properties

 Label 28224bn Number of curves $4$ Conductor $28224$ CM $$\Q(\sqrt{-7})$$ Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("bn1")

E.isogeny_class()

## Elliptic curves in class 28224bn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
28224.dw4 28224bn1 $$[0, 0, 0, -1260, 21168]$$ $$-3375$$ $$-65548320768$$ $$[2]$$ $$16384$$ $$0.78989$$ $$\Gamma_0(N)$$-optimal $$-7$$
28224.dw3 28224bn2 $$[0, 0, 0, -21420, 1206576]$$ $$16581375$$ $$65548320768$$ $$[2]$$ $$32768$$ $$1.1365$$   $$-28$$
28224.dw2 28224bn3 $$[0, 0, 0, -61740, -7260624]$$ $$-3375$$ $$-7711694390034432$$ $$[2]$$ $$114688$$ $$1.7628$$   $$-7$$
28224.dw1 28224bn4 $$[0, 0, 0, -1049580, -413855568]$$ $$16581375$$ $$7711694390034432$$ $$[2]$$ $$229376$$ $$2.1094$$   $$-28$$

## Rank

sage: E.rank()

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

## Complex multiplication

Each elliptic curve in class 28224bn has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-7})$$.

## Modular form28224.2.a.bn

sage: E.q_eigenform(10)

$$q + 4 q^{11} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.