req advice: Fuj-Siem laptop faulty

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I'm in the UK and I've got a Fujitsu Siemens laptop model Amilo D7830, this one: (wrapping url) \KMT_Customer\KMT_Manuals&FileTypeExtension=pdf&FileNameClient=440-n30403-uk.pdf&FileSize=1001910

it's completely dead, nothing happens when I press the power on button, even if the power supply is plugged in. I've tested the PSU
and it reads 20v at the DC plug. At one time when it was plugged in I saw the battery light (page 16 of the manual) flash red for a
couple of minutes and then it stopped and it hasn't come on again since. I could be wrong but my first suspicion is the battery,
which is an internal Fujitsu Siemens Amilo Series Model 755XX5, Battery Pack: 755-4S4000-S1P1. The main body of it is square
measuring 108mm x 108mm. I've had the battery out but it's got 7 connection points all in a tiny row in one corner.
pictures of the battery here:
Does anyone know if I can test this battery and what connections I should read from?
Any other ideas about what could be wrong?
Thanks for any feedback.

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