The curve has been found the algorithms for ray class fields in Magma, together with an explicit equation.
Using various randomised techniques, the following somewhat sparser defining equation has been found:

y^22 + (x+2)*y^21 + (2*x^6+x^5+2*x^4+x^3+2*x+1)*y^19 + (2*x^7+2*x^6+x^5+2*x^4+2*x^3+2*x^2+2*x+2)*y^18
+ (2*x^10+2*x^5+x^4+2*x^3+2*x)*y^17 + (x^12+x^8+x^7+x^5+x^3+2)*y^16
+ (x^12+2*x^11+2*x^10+2*x^8+2*x^5+x^4+x^3+x^2+2*x+1)*y^15+ (2*x^12+2*x^10+x^9+x^8+x^6+x^4+x^2)*y^14
+ (2*x^17+2*x^16+2*x^15+2*x^10+2*x^5+x^3+x^2+x+2)*y^13
+ (2*x^20+x^19+x^17+2*x^15+2*x^14+x^12+2*x^8+x^4+2*x^3+2*x^2+x+1)*y^12
+ (x^22+x^21+x^19+x^18+2*x^17+2*x^12+x^11+x^10+x^9+x^8+x^6+2*x^5+x^3+x^2+x)*y^11
+ (2*x^24+2*x^22+2*x^21+2*x^20+x^19+2*x^18+2*x^17+x^16+x^14+x^13+x^10+2*x^8+2*x^5+2*x^4+x^3+x^2)*y^10
+ (x^25+x^21+x^20+2*x^18+x^17+2*x^15+x^14+x^11+2*x^10+2*x^9+2*x^8+x^7+2*x^5+2*x^4)*y^9
+ (x^28+2*x^23+x^21+2*x^20+2*x^19+x^17+x^14+x^6+x^5)*y^8
+ (2*x^28+2*x^26+2*x^25+x^24+2*x^23+2*x^22+x^21+x^19+x^16+x^15+x^14+x^13+x^11)*y^7
+ (x^32+x^30+2*x^27+x^26+2*x^24+2*x^23+x^22+x^21+x^19+x^16+x^15+x^13+x^11+2*x^10+x^8+2*x^6)*y^6
+ (2*x^34+x^32+x^31+x^30+2*x^29+x^27+x^26+x^25+2*x^24+x^23+x^22+2*x^19+2*x^17+2*x^16+x^15+x^14+2*x^13+2*x^10+x^9+x^8+2*x^7)*y^5
+ (2*x^36+x^35+2*x^33+2*x^32+2*x^31+2*x^30+x^29+2*x^26+x^24+x^23+2*x^22+x^21+x^20+2*x^19+x^17+2*x^16+2*x^15+2*x^14+x^13+2*x^12+2*x^11+2*x^8)*y^4
+ (2*x^38+x^34+x^32+2*x^30+x^27+x^26+x^24+2*x^23+x^21+2*x^20+x^19+2*x^17+2*x^15+x^14+x^13+2*x^12+x^11)*y^3
+ (x^39+x^38+x^34+x^33+x^31+x^29+x^27+2*x^26+2*x^25+x^23+x^22+2*x^21+x^19+2*x^18+x^17+2*x^15)*y^2
+ (x^42+2*x^40+x^38+x^37+2*x^36+x^35+2*x^34+2*x^33+x^32+x^31+2*x^30+2*x^29+2*x^28+2*x^26+x^25+x^23+x^21+2*x^20+x^18+2*x^17)*y
+ 2*x^44+x^43+2*x^42+2*x^41+2*x^38+x^36+x^35+2*x^33+x^31+x^29+x^28+x^26+2*x^23+x^21+x^20

The added reference mainly serves for verification of the data.