# Properties

 Label 30030.p Number of curves 8 Conductor 30030 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("30030.p1")
sage: E.isogeny_class()

## Elliptic curves in class 30030.p

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
30030.p1 30030r8 [1, 0, 1, -105752574, -418583571128] 2 5971968
30030.p2 30030r6 [1, 0, 1, -6887574, -5960607128] 4 2985984
30030.p3 30030r5 [1, 0, 1, -2313909, 426827206] 6 1990656
30030.p4 30030r3 [1, 0, 1, -1887574, 907392872] 2 1492992
30030.p5 30030r2 [1, 0, 1, -1840059, 959624146] 12 995328
30030.p6 30030r1 [1, 0, 1, -1839559, 960172346] 6 497664 $$\Gamma_0(N)$$-optimal
30030.p7 30030r4 [1, 0, 1, -1374209, 1457338286] 6 1990656
30030.p8 30030r7 [1, 0, 1, 11977426, -32869643128] 2 5971968

## Rank

sage: E.rank()

The elliptic curves in class 30030.p have rank $$1$$.

## Modular form 30030.2.a.p

sage: E.q_eigenform(10)
$$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + q^{13} - q^{14} - q^{15} + q^{16} - 6q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.