# Properties

 Label 30912.w Number of curves $6$ Conductor $30912$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("30912.w1")

sage: E.isogeny_class()

## Elliptic curves in class 30912.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
30912.w1 30912bk6 [0, -1, 0, -5064577, 4388596513] [2] 786432
30912.w2 30912bk4 [0, -1, 0, -1133697, -464225247] [2] 393216
30912.w3 30912bk3 [0, -1, 0, -324737, 64914465] [2, 2] 393216
30912.w4 30912bk2 [0, -1, 0, -73857, -6586335] [2, 2] 196608
30912.w5 30912bk1 [0, -1, 0, 8063, -573407] [2] 98304 $$\Gamma_0(N)$$-optimal
30912.w6 30912bk5 [0, -1, 0, 401023, 313269537] [2] 786432

## Rank

sage: E.rank()

The elliptic curves in class 30912.w have rank $$1$$.

## Modular form 30912.2.a.w

sage: E.q_eigenform(10)

$$q - q^{3} + 2q^{5} - q^{7} + q^{9} - 4q^{11} + 2q^{13} - 2q^{15} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.