# Properties

 Label 22320bn Number of curves $6$ Conductor $22320$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("22320.k1")

sage: E.isogeny_class()

## Elliptic curves in class 22320bn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22320.k6 22320bn1 [0, 0, 0, 8637, -1750462] [2] 98304 $$\Gamma_0(N)$$-optimal
22320.k5 22320bn2 [0, 0, 0, -175683, -26781118] [2, 2] 196608
22320.k4 22320bn3 [0, 0, 0, -532803, 116709698] [2] 393216
22320.k2 22320bn4 [0, 0, 0, -2767683, -1772233918] [2, 2] 393216
22320.k3 22320bn5 [0, 0, 0, -2724483, -1830234238] [2] 786432
22320.k1 22320bn6 [0, 0, 0, -44282883, -113423212798] [2] 786432

## Rank

sage: E.rank()

The elliptic curves in class 22320bn have rank $$0$$.

## Modular form 22320.2.a.k

sage: E.q_eigenform(10)

$$q - q^{5} - 4q^{11} + 6q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.