# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 75712z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75712.bw4 75712z1 [0, 0, 0, 676, 35152] [2] 73728 $$\Gamma_0(N)$$-optimal
75712.bw3 75712z2 [0, 0, 0, -12844, 527280] [2, 2] 147456
75712.bw2 75712z3 [0, 0, 0, -39884, -2425488] [2] 294912
75712.bw1 75712z4 [0, 0, 0, -202124, 34976240] [2] 294912

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 75712.2.a.z

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.