# Properties

 Label 75712bs Number of curves $4$ Conductor $75712$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bs1")

sage: E.isogeny_class()

## Elliptic curves in class 75712bs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75712.bv4 75712bs1 [0, 0, 0, 676, -35152] [2] 73728 $$\Gamma_0(N)$$-optimal
75712.bv3 75712bs2 [0, 0, 0, -12844, -527280] [2, 2] 147456
75712.bv2 75712bs3 [0, 0, 0, -39884, 2425488] [2] 294912
75712.bv1 75712bs4 [0, 0, 0, -202124, -34976240] [2] 294912

## Rank

sage: E.rank()

The elliptic curves in class 75712bs have rank $$0$$.

## Complex multiplication

The elliptic curves in class 75712bs do not have complex multiplication.

## Modular form 75712.2.a.bs

sage: E.q_eigenform(10)

$$q + 2q^{5} - q^{7} - 3q^{9} + 4q^{11} - 6q^{17} - 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.