# Properties

 Label 80223c Number of curves $6$ Conductor $80223$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("80223.j1")

sage: E.isogeny_class()

## Elliptic curves in class 80223c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
80223.j4 80223c1 [1, 1, 0, -65221, -6438296] [2] 163840 $$\Gamma_0(N)$$-optimal
80223.j3 80223c2 [1, 1, 0, -65826, -6313545] [2, 2] 327680
80223.j5 80223c3 [1, 1, 0, 26739, -22549446] [2] 655360
80223.j2 80223c4 [1, 1, 0, -168071, 17918520] [2, 2] 655360
80223.j6 80223c5 [1, 1, 0, 468994, 122014941] [2] 1310720
80223.j1 80223c6 [1, 1, 0, -2441056, 1466719159] [2] 1310720

## Rank

sage: E.rank()

The elliptic curves in class 80223c have rank $$0$$.

## Modular form 80223.2.a.j

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - 2q^{5} - q^{6} - 3q^{8} + q^{9} - 2q^{10} + q^{12} - q^{13} + 2q^{15} - q^{16} - q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.