# Properties

 Label 112530bd Number of curves $6$ Conductor $112530$ CM no Rank $2$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("112530.be1")

sage: E.isogeny_class()

## Elliptic curves in class 112530bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
112530.be6 112530bd1 [1, 0, 1, 7257, 1348906] [2] 655360 $$\Gamma_0(N)$$-optimal
112530.be5 112530bd2 [1, 0, 1, -147623, 20615978] [2, 2] 1310720
112530.be4 112530bd3 [1, 0, 1, -447703, -89933494] [2] 2621440
112530.be2 112530bd4 [1, 0, 1, -2325623, 1364877578] [2, 2] 2621440
112530.be3 112530bd5 [1, 0, 1, -2289323, 1409555618] [2] 5242880
112530.be1 112530bd6 [1, 0, 1, -37209923, 87361653938] [2] 5242880

## Rank

sage: E.rank()

The elliptic curves in class 112530bd have rank $$2$$.

## Modular form 112530.2.a.be

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} + q^{12} - 6q^{13} + q^{15} + q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.