# Properties

 Label 121275.cf Number of curves 6 Conductor 121275 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("121275.cf1")

sage: E.isogeny_class()

## Elliptic curves in class 121275.cf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
121275.cf1 121275eq6 [1, -1, 1, -49822205, -135345043578] [2] 7864320
121275.cf2 121275eq4 [1, -1, 1, -3131330, -2089286328] [2, 2] 3932160
121275.cf3 121275eq2 [1, -1, 1, -430205, 60809172] [2, 2] 1966080
121275.cf4 121275eq1 [1, -1, 1, -375080, 88481922] [2] 983040 $$\Gamma_0(N)$$-optimal
121275.cf5 121275eq5 [1, -1, 1, 341545, -6472054578] [2] 7864320
121275.cf6 121275eq3 [1, -1, 1, 1388920, 439187172] [2] 3932160

## Rank

sage: E.rank()

The elliptic curves in class 121275.cf have rank $$0$$.

## Modular form 121275.2.a.cf

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.