first:

the relays you've got can't wired directly to a arduino.

the data sheet isn't doesn't tell what current it draws / peek current not even the coil resistance or the build in resistor in series with the coil.

outgoing form the average power consumption of the coil divided by the max arduino output voltage delivers the lowest average current

P = V * I ==> I = P / U --> I = .29W / 5V = 58mA

the maximum current drawn or shucked to a IO is generally ± 25mA
the absolute maximum generally ±40 - 50mA

maybe the arduino survives the current of 58mA but it can die,
here you can find more info about not exceeding the max current.

also always put a diode across the relay coil to prevent frying the micro controller from the reverse voltage spikes coils generate when they are unloaded

it is not deadly if you don't put the diode, because the arduino outputs are push-pull; which means that that a output is shorted 5V (high) or shorted to gnd(0V) (low). So the coil is never be unloaded / floating. (but for transistors is it needed).

something like this, the transistor can handle more current than a relay. the 12V in your case can variate between 5V and 24V.

the transistor type can be any type where

• max Vce higher is than the supply voltage (the 5 to 24V) (almost any common transistor is can handle that)
• max IC is higher than 100mA
• max Power doesn't care because the transistor will act like a switch

by example the BC547B is an excellent very common transistor and has a gain (HFE) about 200 - 250.

the resistor value can be calculated with ohms law.

R = V / I

• V = (5V -.07V) 5V is the max output voltage of the arduino, 0.7V is the voltage drop in the BE of the transistor.
  
• I = Ic / HFe
  
  • IC = the max current drawn by the relay the previous maths says 58mA but I 'm not sure if more or less. thus i take 100mA
  • HFe is the HFe of the transistor 200

I = 100mA / 200 = .5mA = 500μA

R = 5V / 500μA = 10kΩ

important

the HFe is very dependent from time, temperature, even the collector current and base current may affect the HFe.

so to be sure mostly the resistor value is divided by 2 or more.

but because we've taken the current almost twice as high, and the minimum HFe (the BC547b's HFe is between 200 and 450) and it is not a disaster is the transistor is not fully saturated but maybe 90% (VCE about 1V in stead of 0.3V) and 10kΩ a very common resistor we can take 10kΩ but if you want to be really safe, you can take a 5.6kΩ or less.

Second:

multiplexing

there are several ways to multiplex
very common is matrixing:

just replace every led by a relay

how it works

first select a column than the row

the image is for demonstrating only, don't care about pin numbers

connect each column pin and row trough a transistor / transistor chip to the arduino if you want 48 relays then a 6 by 8 matrix may do the trick

there exist ICs that can do something similar

More IO

1. there exist io expanders over I2C or SPI
   with a lot of libraries for arduino
   
   generally a io expander acts like a io on the arduino and can handle about the same current maybe more or less

2. bcd to decimal converts a binary number to a single high output
   
   the 74HC42 (by example) can do this
   I don't know if there exist a library for a arduino. but if necessary, I can write you a code example that works (or a library is also possible)

   3. shift registers explained on the arduino site shiftout()
      
      the 74hc595 can do this

You were applying the DC1 specs to your AC load application by mistake.

The life cycle rating is limited by surge starting current and the breaking switch off current more than the steady state current of 10A. It is the energy dissipated across the contacts that matters or the V*I product during switching.

But in DC1 the curves are significantly derated becuase the breaking current is always the same as the steady current for "pure" resistive loads.

In AC loads with sinusoidal current most of the time it is less than the peak current so the average temperature rise is lower on the contacts, which controls life cycle time.

For conservative design, increase the current rating.