# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("30030.bt1")
sage: E.isogeny_class()

## Elliptic curves in class 30030bt

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
30030.bt7 30030bt1 [1, 0, 0, -749461, 263897441] 12 774144 $$\Gamma_0(N)$$-optimal
30030.bt6 30030bt2 [1, 0, 0, -12180181, 16360637345] 12 1548288
30030.bt8 30030bt3 [1, 0, 0, 4220699, 295790705] 4 2322432
30030.bt5 30030bt4 [1, 0, 0, -12369181, 15826636745] 6 3096576
30030.bt2 30030bt5 [1, 0, 0, -194882701, 1047131714681] 6 3096576
30030.bt4 30030bt6 [1, 0, 0, -16956121, 2366883701] 4 4644864
30030.bt3 30030bt7 [1, 0, 0, -177737371, -908394585049] 2 9289728
30030.bt1 30030bt8 [1, 0, 0, -195003991, 1045763011475] 2 9289728

## Rank

sage: E.rank()

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

## Modular form 30030.2.a.bt

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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.