# Properties

 Label 76230.ea Number of curves $8$ Conductor $76230$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("76230.ea1")

sage: E.isogeny_class()

## Elliptic curves in class 76230.ea

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
76230.ea1 76230ek8 [1, -1, 1, -7025162, 4510720811] [2] 6635520
76230.ea2 76230ek5 [1, -1, 1, -6273752, 6049948259] [2] 2211840
76230.ea3 76230ek6 [1, -1, 1, -2941412, -1889332189] [2, 2] 3317760
76230.ea4 76230ek3 [1, -1, 1, -2919632, -1919440861] [2] 1658880
76230.ea5 76230ek2 [1, -1, 1, -393152, 94076579] [2, 2] 1105920
76230.ea6 76230ek4 [1, -1, 1, -88232, 236047331] [2] 2211840
76230.ea7 76230ek1 [1, -1, 1, -44672, -1267549] [2] 552960 $$\Gamma_0(N)$$-optimal
76230.ea8 76230ek7 [1, -1, 1, 793858, -6362691541] [2] 6635520

## Rank

sage: E.rank()

The elliptic curves in class 76230.ea have rank $$1$$.

## Modular form 76230.2.a.ea

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.