# Properties

 Label 22848.cp Number of curves $4$ Conductor $22848$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("cp1")

sage: E.isogeny_class()

## Elliptic curves in class 22848.cp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
22848.cp1 22848bc4 $$[0, 1, 0, -325057, 71224223]$$ $$14489843500598257/6246072$$ $$1637370298368$$ $$[4]$$ $$147456$$ $$1.6863$$
22848.cp2 22848bc3 $$[0, 1, 0, -43457, -1857633]$$ $$34623662831857/14438442312$$ $$3784951021436928$$ $$[2]$$ $$147456$$ $$1.6863$$
22848.cp3 22848bc2 $$[0, 1, 0, -20417, 1096095]$$ $$3590714269297/73410624$$ $$19244154617856$$ $$[2, 2]$$ $$73728$$ $$1.3397$$
22848.cp4 22848bc1 $$[0, 1, 0, 63, 51615]$$ $$103823/4386816$$ $$-1149977493504$$ $$[2]$$ $$36864$$ $$0.99312$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 22848.cp have rank $$1$$.

## Complex multiplication

The elliptic curves in class 22848.cp do not have complex multiplication.

## Modular form 22848.2.a.cp

sage: E.q_eigenform(10)

$$q + q^{3} + 2q^{5} - q^{7} + q^{9} + 6q^{13} + 2q^{15} + q^{17} + O(q^{20})$$

## 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.