# Properties

 Label 127050.gr Number of curves $8$ Conductor $127050$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("127050.gr1")

sage: E.isogeny_class()

## Elliptic curves in class 127050.gr

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
127050.gr1 127050gb8 [1, 1, 1, -19514338, -20882966719] [2] 19906560
127050.gr2 127050gb5 [1, 1, 1, -17427088, -28009019719] [2] 6635520
127050.gr3 127050gb6 [1, 1, 1, -8170588, 8746908281] [2, 2] 9953280
127050.gr4 127050gb3 [1, 1, 1, -8110088, 8886300281] [2] 4976640
127050.gr5 127050gb2 [1, 1, 1, -1092088, -435539719] [2, 2] 3317760
127050.gr6 127050gb4 [1, 1, 1, -245088, -1092811719] [2] 6635520
127050.gr7 127050gb1 [1, 1, 1, -124088, 5868281] [2] 1658880 $$\Gamma_0(N)$$-optimal
127050.gr8 127050gb7 [1, 1, 1, 2205162, 29456905281] [2] 19906560

## Rank

sage: E.rank()

The elliptic curves in class 127050.gr have rank $$0$$.

## Modular form 127050.2.a.gr

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} + q^{9} - q^{12} + 2q^{13} + q^{14} + q^{16} - 6q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.