# Properties

 Label 47040.hb Number of curves $6$ Conductor $47040$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("hb1")

sage: E.isogeny_class()

## Elliptic curves in class 47040.hb

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.hb1 47040hc6 [0, 1, 0, -52684865, -147207093825] [2] 2359296
47040.hb2 47040hc4 [0, 1, 0, -3292865, -2300844225] [2, 2] 1179648
47040.hb3 47040hc5 [0, 1, 0, -3073345, -2620597057] [4] 2359296
47040.hb4 47040hc3 [0, 1, 0, -1160385, 454345023] [2] 1179648
47040.hb5 47040hc2 [0, 1, 0, -219585, -30919617] [2, 2] 589824
47040.hb6 47040hc1 [0, 1, 0, 31295, -2971585] [2] 294912 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 47040.hb have rank $$1$$.

## Complex multiplication

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

## Modular form 47040.2.a.hb

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} + q^{9} + 4q^{11} - 2q^{13} + q^{15} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.