# Properties

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

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("bw1")

sage: E.isogeny_class()

## Elliptic curves in class 75712.bw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75712.bw1 75712z4 $$[0, 0, 0, -202124, 34976240]$$ $$1443468546/7$$ $$4428616564736$$ $$[2]$$ $$294912$$ $$1.6270$$
75712.bw2 75712z3 $$[0, 0, 0, -39884, -2425488]$$ $$11090466/2401$$ $$1519015481704448$$ $$[2]$$ $$294912$$ $$1.6270$$
75712.bw3 75712z2 $$[0, 0, 0, -12844, 527280]$$ $$740772/49$$ $$15500157976576$$ $$[2, 2]$$ $$147456$$ $$1.2804$$
75712.bw4 75712z1 $$[0, 0, 0, 676, 35152]$$ $$432/7$$ $$-553577070592$$ $$[2]$$ $$73728$$ $$0.93387$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 75712.2.a.bw

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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.