# Properties

 Label 121275.bh Number of curves $6$ Conductor $121275$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("121275.bh1")

sage: E.isogeny_class()

## Elliptic curves in class 121275.bh

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
121275.bh1 121275dh6 [1, -1, 1, -146081480, -679511326728] [2] 18874368
121275.bh2 121275dh4 [1, -1, 1, -9647105, -9345676728] [2, 2] 9437184
121275.bh3 121275dh2 [1, -1, 1, -2976980, 1846793022] [2, 2] 4718592
121275.bh4 121275dh1 [1, -1, 1, -2921855, 1923086022] [4] 2359296 $$\Gamma_0(N)$$-optimal
121275.bh5 121275dh3 [1, -1, 1, 2811145, 8155849272] [2] 9437184
121275.bh6 121275dh5 [1, -1, 1, 20065270, -55578132228] [2] 18874368

## Rank

sage: E.rank()

The elliptic curves in class 121275.bh have rank $$1$$.

## Modular form 121275.2.a.bh

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + 3q^{8} - q^{11} - 2q^{13} - q^{16} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.