IPhone 5s says E instead of LTE?

OK so I just bought an iPhone 5s off of my friend who originally got the phone from Apple and chose Verizon as a carrier there. I was told the phone would work on T-Mobile's network. Which was true because the new SIM card I got from T-Mobile worked and I was connected to the network. The only problem is is that I'm supposed to have 4G LTE data and it comes up as E. I went to T-Mobile and they couldn't figure it out. The APN was the same. They told me to go to Apple, who then told me to go to Verizon, because they think the reason why it isn't working is because the "internet" isn't unlocked. Apple told me to go to Verizon to have them unlock it. I went to Verizon and they told me that they don't unlock/had the equipment and that they are "even surprised that the phone works and makes calls" which was BS because it CLEARLY works. Apple told me it's fully possible to get the LTE. I just feel that Verizon cut me short because I don't have business with them. Everywhere I look says that Verizon can unlock phones. Should I have my friend call tech-support through Verizon? Who's telling the truth? What do I do?

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