# Properties

 Label 47040.dp Number of curves $4$ Conductor $47040$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("dp1")

sage: E.isogeny_class()

## Elliptic curves in class 47040.dp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
47040.dp1 47040fi4 $$[0, -1, 0, -219585, 39678465]$$ $$303735479048/105$$ $$404787855360$$ $$[4]$$ $$294912$$ $$1.5831$$
47040.dp2 47040fi2 $$[0, -1, 0, -13785, 617625]$$ $$601211584/11025$$ $$5312840601600$$ $$[2, 2]$$ $$147456$$ $$1.2365$$
47040.dp3 47040fi1 $$[0, -1, 0, -1780, -13838]$$ $$82881856/36015$$ $$271176239040$$ $$[2]$$ $$73728$$ $$0.88997$$ $$\Gamma_0(N)$$-optimal
47040.dp4 47040fi3 $$[0, -1, 0, -65, 1778337]$$ $$-8/354375$$ $$-1366159011840000$$ $$[2]$$ $$294912$$ $$1.5831$$

## Rank

sage: E.rank()

The elliptic curves in class 47040.dp have rank $$0$$.

## Complex multiplication

The elliptic curves in class 47040.dp do not have complex multiplication.

## Modular form 47040.2.a.dp

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{9} + 4q^{11} + 6q^{13} - q^{15} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.