# Properties

 Label 75712.bv Number of curves $4$ Conductor $75712$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bv1")

sage: E.isogeny_class()

## Elliptic curves in class 75712.bv

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75712.bv1 75712bs4 $$[0, 0, 0, -202124, -34976240]$$ $$1443468546/7$$ $$4428616564736$$ $$$$ $$294912$$ $$1.6270$$
75712.bv2 75712bs3 $$[0, 0, 0, -39884, 2425488]$$ $$11090466/2401$$ $$1519015481704448$$ $$$$ $$294912$$ $$1.6270$$
75712.bv3 75712bs2 $$[0, 0, 0, -12844, -527280]$$ $$740772/49$$ $$15500157976576$$ $$[2, 2]$$ $$147456$$ $$1.2804$$
75712.bv4 75712bs1 $$[0, 0, 0, 676, -35152]$$ $$432/7$$ $$-553577070592$$ $$$$ $$73728$$ $$0.93387$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 75712.2.a.bv

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. 