# Properties

 Label 90.c Number of curves $8$ Conductor $90$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("90.c1")

sage: E.isogeny_class()

## Elliptic curves in class 90.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
90.c1 90c8 [1, -1, 1, -48002, 4059929] [6] 192
90.c2 90c7 [1, -1, 1, -4082, 14681] [6] 192
90.c3 90c6 [1, -1, 1, -3002, 63929] [2, 6] 96
90.c4 90c4 [1, -1, 1, -2597, -50281] [2] 64
90.c5 90c5 [1, -1, 1, -617, 5231] [2] 64
90.c6 90c2 [1, -1, 1, -167, -709] [2, 2] 32
90.c7 90c3 [1, -1, 1, -122, 1721] [12] 48
90.c8 90c1 [1, -1, 1, 13, -61] [4] 16 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 90.c have rank $$0$$.

## Modular form90.2.a.c

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{5} - 4q^{7} + q^{8} + q^{10} + 2q^{13} - 4q^{14} + q^{16} - 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.