# Properties

 Label 162240.hj Number of curves $6$ Conductor $162240$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("162240.hj1")

sage: E.isogeny_class()

## Elliptic curves in class 162240.hj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
162240.hj1 162240fc6 [0, 1, 0, -97506465, -370626330177] [2] 16515072
162240.hj2 162240fc3 [0, 1, 0, -9139745, 10623846975] [2] 8257536
162240.hj3 162240fc4 [0, 1, 0, -6111265, -5758412737] [2, 2] 8257536
162240.hj4 162240fc5 [0, 1, 0, -1244065, -14674149697] [2] 16515072
162240.hj5 162240fc2 [0, 1, 0, -703265, 83308863] [2, 2] 4128768
162240.hj6 162240fc1 [0, 1, 0, 162015, 10106175] [2] 2064384 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 162240.hj have rank $$1$$.

## Modular form 162240.2.a.hj

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.