They come in all different colours and shades: pearly white, bright red, cloudy-grey, dark blue, gleaming golden yellow.

But it is the curvature of the fish that accounts for 60 per cent of the final score, explained competition organiser Isamu Hattori, who runs Japan’s main association for breeders of koi carp.

Colour and contrast make up another 30 per cent, he said.

And the final 10 per cent? “Hinkaku” – a concept that is tricky to define and even harder to judge, best translated as the “presence” or “aura” of the fish.

“Hinkaku. It’s either there in the genes at birth, or it’s not,” mused Mikinori Kurikara, a koi breeder in Saitama, north of Tokyo, who says he can spot it in fish when they are eight or nine months old.

“Put it this way, it’s like looking after your own children every day. You care for your kids and want them to grow healthy. In the same way, you take care of these fish, appreciate them and adore them.”

At his farm, thousands of tiny nishikigoi dart around deep basins of carefully purified water, meticulously divided by age and colour.

A less glorious fate awaits the other koi who have not been fortunate enough to catch the eye of the breeder: they are sold as feed for tropical fish.

“It’s a really delicate job, really difficult. Everything matters: the ground, the water quality, the food,” explained the 48-year-old, who took over the farm from his father and is training his son, half his age, in the subtle arts of koi breeding. “We have many secrets. But even if we let them slip, it wouldn’t work. You have to be able to feel it.”

These days, any self-respecting traditional Japanese garden has plenty of colourful koi gracing its ponds, but it is a relatively recent tradition.

Around 200 years ago, villagers in the mountainous region around Niigata started to practise genetic engineering without knowing what they were doing.

For the first time, they began to cross-breed rare colourful carp, not for food but for pure aesthetic value.

The craze for nishikigoi gradually took over the whole of Japan and then spread to other parts of Asia.

They are especially popular in China, where carp swimming against the tide symbolises the idea of perseverance leading to riches – rather like people climbing the social ladder, said Yutaka Suga, professor at the Institute for Advanced Studies on Asia at Tokyo University.

Today, koi is big business and Japanese exports are booming – 90 per cent of domestic production is exported and sold at auction.

In 2016, Japan exported a record 295 tonnes of koi carp, generating turnover of 3.5 billion yen (US$31 million), an increase of almost 50 per cent from 2007, according to Japan’s agriculture ministry.

As for individual carp, “the prices have become insane” said Hattori.

“Today, a two-year-old carp can sell for 30 million yen each (US$265,000) whereas 10 years ago, 2 million yen was already a very good price,” he said.

Like racehorse owners, many foreigners leave their prized possessions in their home – Japanese farms – so they can compete in the most prestigious fishy pageants, which are only open to domestic rearers.

One such owner, Chinese koi collector Yuan Jiandong, was in Tokyo to cheer on some of his own carp.

“It’s not a way of making money. It’s a way of spending it for fun,” laughed the pharmaceutical boss from Shanghai.

But owning koi is so much more than a vulgar display of wealth, he said.

“When you see these beautiful fish gliding around in your pond, you forget the stresses of daily life and you find peace of mind.”

And you can’t put a price on that.