LMFDB - Number field 32.0.44580746322432044939368312527587514829553937678336.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 2*x^30 + 9*x^28 + 4*x^26 + 26*x^24 - 45*x^22 + 10*x^20 - 403*x^18 - 171*x^16 - 1612*x^14 + 160*x^12 - 2880*x^10 + 6656*x^8 + 4096*x^6 + 36864*x^4 + 32768*x^2 + 65536)

gp: K = bnfinit(y^32 + 2*y^30 + 9*y^28 + 4*y^26 + 26*y^24 - 45*y^22 + 10*y^20 - 403*y^18 - 171*y^16 - 1612*y^14 + 160*y^12 - 2880*y^10 + 6656*y^8 + 4096*y^6 + 36864*y^4 + 32768*y^2 + 65536, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 2*x^30 + 9*x^28 + 4*x^26 + 26*x^24 - 45*x^22 + 10*x^20 - 403*x^18 - 171*x^16 - 1612*x^14 + 160*x^12 - 2880*x^10 + 6656*x^8 + 4096*x^6 + 36864*x^4 + 32768*x^2 + 65536);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 2*x^30 + 9*x^28 + 4*x^26 + 26*x^24 - 45*x^22 + 10*x^20 - 403*x^18 - 171*x^16 - 1612*x^14 + 160*x^12 - 2880*x^10 + 6656*x^8 + 4096*x^6 + 36864*x^4 + 32768*x^2 + 65536)

$ x^{32} + 2 x^{30} + 9 x^{28} + 4 x^{26} + 26 x^{24} - 45 x^{22} + 10 x^{20} - 403 x^{18} - 171 x^{16} + \cdots + 65536 $ x^32 + 2*x^30 + 9*x^28 + 4*x^26 + 26*x^24 - 45*x^22 + 10*x^20 - 403*x^18 - 171*x^16 - 1612*x^14 + 160*x^12 - 2880*x^10 + 6656*x^8 + 4096*x^6 + 36864*x^4 + 32768*x^2 + 65536

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$32$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 16]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$44580746322432044939368312527587514829553937678336$ 44580746322432044939368312527587514829553937678336 $\medspace = 2^{32}\cdot 3^{24}\cdot 61^{16}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$35.61$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$2\cdot 3^{3/4}61^{2/3}\approx 70.6465286301374$
Ramified primes:	$2$, $3$, $61$ 2, 3, 61	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{32768}$

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{6}+\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{7}+\frac{1}{3}a$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{6}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{7}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{14}+\frac{1}{9}a^{12}+\frac{1}{9}a^{10}+\frac{1}{9}a^{8}+\frac{1}{9}a^{6}-\frac{2}{9}a^{4}-\frac{2}{9}a^{2}-\frac{2}{9}$, $\frac{1}{18}a^{17}-\frac{1}{9}a^{15}+\frac{1}{18}a^{13}-\frac{1}{9}a^{11}-\frac{1}{9}a^{9}-\frac{5}{18}a^{7}-\frac{4}{9}a^{5}-\frac{5}{18}a^{3}-\frac{5}{18}a$, $\frac{1}{36}a^{18}-\frac{1}{18}a^{16}+\frac{1}{36}a^{14}+\frac{1}{9}a^{12}-\frac{1}{18}a^{10}-\frac{5}{36}a^{8}-\frac{1}{18}a^{6}+\frac{13}{36}a^{4}+\frac{13}{36}a^{2}-\frac{1}{3}$, $\frac{1}{72}a^{19}-\frac{1}{36}a^{17}+\frac{1}{72}a^{15}-\frac{1}{9}a^{13}+\frac{5}{36}a^{11}-\frac{5}{72}a^{9}-\frac{13}{36}a^{7}-\frac{35}{72}a^{5}-\frac{23}{72}a^{3}+\frac{1}{3}a$, $\frac{1}{144}a^{20}-\frac{1}{72}a^{18}+\frac{1}{144}a^{16}+\frac{1}{9}a^{14}+\frac{5}{72}a^{12}-\frac{5}{144}a^{10}-\frac{1}{72}a^{8}+\frac{37}{144}a^{6}+\frac{49}{144}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{288}a^{21}-\frac{1}{144}a^{19}+\frac{1}{288}a^{17}-\frac{1}{9}a^{15}+\frac{5}{144}a^{13}+\frac{43}{288}a^{11}-\frac{1}{144}a^{9}-\frac{59}{288}a^{7}+\frac{1}{288}a^{5}+\frac{1}{3}a^{3}-\frac{1}{6}a$, $\frac{1}{1728}a^{22}-\frac{1}{864}a^{20}+\frac{1}{1728}a^{18}-\frac{1}{18}a^{16}+\frac{23}{288}a^{14}-\frac{71}{576}a^{12}+\frac{7}{96}a^{10}+\frac{29}{192}a^{8}+\frac{89}{192}a^{6}-\frac{17}{54}a^{4}+\frac{41}{108}a^{2}-\frac{13}{27}$, $\frac{1}{3456}a^{23}-\frac{1}{1728}a^{21}+\frac{1}{3456}a^{19}-\frac{1}{36}a^{17}+\frac{23}{576}a^{15}-\frac{71}{1152}a^{13}-\frac{25}{192}a^{11}-\frac{35}{384}a^{9}+\frac{25}{384}a^{7}-\frac{53}{108}a^{5}-\frac{31}{216}a^{3}-\frac{2}{27}a$, $\frac{1}{6912}a^{24}-\frac{1}{3456}a^{22}+\frac{1}{6912}a^{20}-\frac{1}{72}a^{18}+\frac{23}{1152}a^{16}-\frac{71}{2304}a^{14}+\frac{13}{128}a^{12}-\frac{35}{768}a^{10}+\frac{25}{768}a^{8}-\frac{17}{216}a^{6}+\frac{185}{432}a^{4}-\frac{1}{27}a^{2}-\frac{1}{3}$, $\frac{1}{13824}a^{25}-\frac{1}{6912}a^{23}+\frac{1}{13824}a^{21}-\frac{1}{144}a^{19}+\frac{23}{2304}a^{17}-\frac{71}{4608}a^{15}+\frac{13}{256}a^{13}+\frac{221}{1536}a^{11}+\frac{25}{1536}a^{9}-\frac{89}{432}a^{7}+\frac{41}{864}a^{5}+\frac{13}{27}a^{3}-\frac{1}{2}a$, $\frac{1}{27648}a^{26}-\frac{1}{13824}a^{24}+\frac{1}{27648}a^{22}-\frac{1}{288}a^{20}+\frac{23}{4608}a^{18}-\frac{71}{9216}a^{16}-\frac{217}{1536}a^{14}+\frac{221}{3072}a^{12}-\frac{487}{3072}a^{10}-\frac{89}{864}a^{8}-\frac{823}{1728}a^{6}-\frac{5}{54}a^{4}-\frac{1}{12}a^{2}$, $\frac{1}{55296}a^{27}-\frac{1}{27648}a^{25}+\frac{1}{55296}a^{23}-\frac{1}{576}a^{21}+\frac{23}{9216}a^{19}-\frac{71}{18432}a^{17}-\frac{217}{3072}a^{15}-\frac{803}{6144}a^{13}+\frac{179}{2048}a^{11}-\frac{89}{1728}a^{9}-\frac{247}{3456}a^{7}-\frac{23}{108}a^{5}-\frac{1}{24}a^{3}-\frac{1}{2}a$, $\frac{1}{110592}a^{28}-\frac{1}{55296}a^{26}+\frac{1}{110592}a^{24}-\frac{1}{3456}a^{22}+\frac{5}{55296}a^{20}-\frac{149}{110592}a^{18}-\frac{217}{6144}a^{16}+\frac{287}{4096}a^{14}-\frac{295}{12288}a^{12}+\frac{355}{3456}a^{10}-\frac{1123}{6912}a^{8}-\frac{439}{1728}a^{6}+\frac{23}{432}a^{4}+\frac{19}{54}a^{2}+\frac{2}{27}$, $\frac{1}{221184}a^{29}-\frac{1}{110592}a^{27}+\frac{1}{221184}a^{25}-\frac{1}{6912}a^{23}+\frac{5}{110592}a^{21}-\frac{149}{221184}a^{19}-\frac{217}{12288}a^{17}+\frac{287}{8192}a^{15}-\frac{295}{24576}a^{13}-\frac{797}{6912}a^{11}-\frac{1123}{13824}a^{9}+\frac{137}{3456}a^{7}+\frac{167}{864}a^{5}+\frac{19}{108}a^{3}+\frac{10}{27}a$, $\frac{1}{269402112}a^{30}+\frac{7}{6414336}a^{28}-\frac{613}{89800704}a^{26}-\frac{193}{4810752}a^{24}-\frac{881}{4988928}a^{22}-\frac{5493}{3325952}a^{20}-\frac{1088293}{134701056}a^{18}-\frac{248083}{29933568}a^{16}+\frac{704779}{9977856}a^{14}-\frac{778427}{33675264}a^{12}+\frac{35941}{5612544}a^{10}-\frac{32171}{1403136}a^{8}-\frac{73}{150336}a^{6}+\frac{533}{14616}a^{4}-\frac{367}{1044}a^{2}+\frac{3971}{16443}$, $\frac{1}{538804224}a^{31}+\frac{7}{12828672}a^{29}-\frac{613}{179601408}a^{27}-\frac{193}{9621504}a^{25}-\frac{881}{9977856}a^{23}-\frac{5493}{6651904}a^{21}-\frac{1088293}{269402112}a^{19}-\frac{248083}{59867136}a^{17}-\frac{2621173}{19955712}a^{15}+\frac{10446661}{67350528}a^{13}+\frac{35941}{11225088}a^{11}+\frac{435541}{2806272}a^{9}-\frac{50185}{300672}a^{7}+\frac{533}{29232}a^{5}+\frac{677}{2088}a^{3}+\frac{3971}{32886}a$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/3*a^8 - 1/3*a^6 - 1/3*a^2 + 1/3, 1/3*a^9 - 1/3*a^7 - 1/3*a^3 + 1/3*a, 1/3*a^10 - 1/3*a^6 - 1/3*a^4 + 1/3, 1/3*a^11 - 1/3*a^7 - 1/3*a^5 + 1/3*a, 1/3*a^12 + 1/3*a^6 + 1/3, 1/3*a^13 + 1/3*a^7 + 1/3*a, 1/3*a^14 + 1/3*a^6 - 1/3*a^2 - 1/3, 1/3*a^15 + 1/3*a^7 - 1/3*a^3 - 1/3*a, 1/9*a^16 + 1/9*a^14 + 1/9*a^12 + 1/9*a^10 + 1/9*a^8 + 1/9*a^6 - 2/9*a^4 - 2/9*a^2 - 2/9, 1/18*a^17 - 1/9*a^15 + 1/18*a^13 - 1/9*a^11 - 1/9*a^9 - 5/18*a^7 - 4/9*a^5 - 5/18*a^3 - 5/18*a, 1/36*a^18 - 1/18*a^16 + 1/36*a^14 + 1/9*a^12 - 1/18*a^10 - 5/36*a^8 - 1/18*a^6 + 13/36*a^4 + 13/36*a^2 - 1/3, 1/72*a^19 - 1/36*a^17 + 1/72*a^15 - 1/9*a^13 + 5/36*a^11 - 5/72*a^9 - 13/36*a^7 - 35/72*a^5 - 23/72*a^3 + 1/3*a, 1/144*a^20 - 1/72*a^18 + 1/144*a^16 + 1/9*a^14 + 5/72*a^12 - 5/144*a^10 - 1/72*a^8 + 37/144*a^6 + 49/144*a^4 + 1/3*a^2, 1/288*a^21 - 1/144*a^19 + 1/288*a^17 - 1/9*a^15 + 5/144*a^13 + 43/288*a^11 - 1/144*a^9 - 59/288*a^7 + 1/288*a^5 + 1/3*a^3 - 1/6*a, 1/1728*a^22 - 1/864*a^20 + 1/1728*a^18 - 1/18*a^16 + 23/288*a^14 - 71/576*a^12 + 7/96*a^10 + 29/192*a^8 + 89/192*a^6 - 17/54*a^4 + 41/108*a^2 - 13/27, 1/3456*a^23 - 1/1728*a^21 + 1/3456*a^19 - 1/36*a^17 + 23/576*a^15 - 71/1152*a^13 - 25/192*a^11 - 35/384*a^9 + 25/384*a^7 - 53/108*a^5 - 31/216*a^3 - 2/27*a, 1/6912*a^24 - 1/3456*a^22 + 1/6912*a^20 - 1/72*a^18 + 23/1152*a^16 - 71/2304*a^14 + 13/128*a^12 - 35/768*a^10 + 25/768*a^8 - 17/216*a^6 + 185/432*a^4 - 1/27*a^2 - 1/3, 1/13824*a^25 - 1/6912*a^23 + 1/13824*a^21 - 1/144*a^19 + 23/2304*a^17 - 71/4608*a^15 + 13/256*a^13 + 221/1536*a^11 + 25/1536*a^9 - 89/432*a^7 + 41/864*a^5 + 13/27*a^3 - 1/2*a, 1/27648*a^26 - 1/13824*a^24 + 1/27648*a^22 - 1/288*a^20 + 23/4608*a^18 - 71/9216*a^16 - 217/1536*a^14 + 221/3072*a^12 - 487/3072*a^10 - 89/864*a^8 - 823/1728*a^6 - 5/54*a^4 - 1/12*a^2, 1/55296*a^27 - 1/27648*a^25 + 1/55296*a^23 - 1/576*a^21 + 23/9216*a^19 - 71/18432*a^17 - 217/3072*a^15 - 803/6144*a^13 + 179/2048*a^11 - 89/1728*a^9 - 247/3456*a^7 - 23/108*a^5 - 1/24*a^3 - 1/2*a, 1/110592*a^28 - 1/55296*a^26 + 1/110592*a^24 - 1/3456*a^22 + 5/55296*a^20 - 149/110592*a^18 - 217/6144*a^16 + 287/4096*a^14 - 295/12288*a^12 + 355/3456*a^10 - 1123/6912*a^8 - 439/1728*a^6 + 23/432*a^4 + 19/54*a^2 + 2/27, 1/221184*a^29 - 1/110592*a^27 + 1/221184*a^25 - 1/6912*a^23 + 5/110592*a^21 - 149/221184*a^19 - 217/12288*a^17 + 287/8192*a^15 - 295/24576*a^13 - 797/6912*a^11 - 1123/13824*a^9 + 137/3456*a^7 + 167/864*a^5 + 19/108*a^3 + 10/27*a, 1/269402112*a^30 + 7/6414336*a^28 - 613/89800704*a^26 - 193/4810752*a^24 - 881/4988928*a^22 - 5493/3325952*a^20 - 1088293/134701056*a^18 - 248083/29933568*a^16 + 704779/9977856*a^14 - 778427/33675264*a^12 + 35941/5612544*a^10 - 32171/1403136*a^8 - 73/150336*a^6 + 533/14616*a^4 - 367/1044*a^2 + 3971/16443, 1/538804224*a^31 + 7/12828672*a^29 - 613/179601408*a^27 - 193/9621504*a^25 - 881/9977856*a^23 - 5493/6651904*a^21 - 1088293/269402112*a^19 - 248083/59867136*a^17 - 2621173/19955712*a^15 + 10446661/67350528*a^13 + 35941/11225088*a^11 + 435541/2806272*a^9 - 50185/300672*a^7 + 533/29232*a^5 + 677/2088*a^3 + 3971/32886*a

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$, $3$

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$15$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ \frac{3713}{44900352} a^{31} + \frac{259}{801792} a^{29} + \frac{16141}{44900352} a^{27} + \frac{491}{356352} a^{25} - \frac{27499}{22450176} a^{23} + \frac{123119}{44900352} a^{21} - \frac{205325}{11225088} a^{19} + \frac{1051}{14966784} a^{17} - \frac{1341583}{14966784} a^{15} + \frac{390101}{22450176} a^{13} - \frac{696347}{2806272} a^{11} + \frac{125497}{350784} a^{9} - \frac{7705}{33408} a^{7} + \frac{92447}{43848} a^{5} + \frac{1279}{1566} a^{3} + \frac{31519}{5481} a $ (3713)/(44900352)a^(31) + (259)/(801792)a^(29) + (16141)/(44900352)a^(27) + (491)/(356352)a^(25) - (27499)/(22450176)a^(23) + (123119)/(44900352)a^(21) - (205325)/(11225088)a^(19) + (1051)/(14966784)a^(17) - (1341583)/(14966784)a^(15) + (390101)/(22450176)a^(13) - (696347)/(2806272)a^(11) + (125497)/(350784)a^(9) - (7705)/(33408)a^(7) + (92447)/(43848)a^(5) + (1279)/(1566)a^(3) + (31519)/(5481)a (order $12$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$\frac{3713}{44900352}a^{31}+\frac{259}{801792}a^{29}+\frac{16141}{44900352}a^{27}+\frac{491}{356352}a^{25}-\frac{27499}{22450176}a^{23}+\frac{123119}{44900352}a^{21}-\frac{205325}{11225088}a^{19}+\frac{1051}{14966784}a^{17}-\frac{1341583}{14966784}a^{15}+\frac{390101}{22450176}a^{13}-\frac{696347}{2806272}a^{11}+\frac{125497}{350784}a^{9}-\frac{7705}{33408}a^{7}+\frac{92447}{43848}a^{5}+\frac{1279}{1566}a^{3}+\frac{31519}{5481}a-1$, $\frac{50893}{269402112}a^{30}+\frac{1117}{6414336}a^{28}+\frac{37229}{29933568}a^{26}-\frac{1751}{9621504}a^{24}+\frac{108107}{44900352}a^{22}-\frac{69499}{9977856}a^{20}-\frac{90547}{134701056}a^{18}-\frac{1444543}{29933568}a^{16}-\frac{790927}{29933568}a^{14}-\frac{9107617}{67350528}a^{12}+\frac{71059}{5612544}a^{10}-\frac{16067}{155904}a^{8}+\frac{101327}{150336}a^{6}+\frac{118913}{87696}a^{4}+\frac{2735}{1044}a^{2}+\frac{84104}{16443}$, $\frac{1241}{7483392}a^{30}-\frac{803}{1069056}a^{28}+\frac{32261}{22450176}a^{26}-\frac{5845}{1069056}a^{24}+\frac{117083}{11225088}a^{22}-\frac{622273}{22450176}a^{20}+\frac{1307767}{22450176}a^{18}-\frac{822421}{7483392}a^{16}+\frac{336703}{1247232}a^{14}-\frac{1014073}{2494464}a^{12}+\frac{299479}{311808}a^{10}-\frac{1984253}{1403136}a^{8}+\frac{22423}{8352}a^{6}-\frac{95407}{21924}a^{4}+\frac{13675}{3132}a^{2}-\frac{57856}{5481}$, $\frac{41}{442368}a^{30}-\frac{229}{221184}a^{28}+\frac{505}{442368}a^{26}-\frac{137}{18432}a^{24}+\frac{2749}{221184}a^{22}-\frac{14605}{442368}a^{20}+\frac{17795}{221184}a^{18}-\frac{5899}{49152}a^{16}+\frac{18665}{49152}a^{14}-\frac{25303}{55296}a^{12}+\frac{36347}{27648}a^{10}-\frac{13699}{6912}a^{8}+\frac{233}{72}a^{6}-\frac{2993}{432}a^{4}+\frac{421}{108}a^{2}-\frac{469}{27}$, $\frac{87389}{269402112}a^{30}+\frac{1505}{6414336}a^{28}+\frac{154087}{89800704}a^{26}-\frac{6169}{9621504}a^{24}+\frac{67609}{14966784}a^{22}-\frac{36713}{3325952}a^{20}+\frac{48745}{134701056}a^{18}-\frac{691573}{9977856}a^{16}-\frac{530263}{29933568}a^{14}-\frac{14639735}{67350528}a^{12}+\frac{347105}{5612544}a^{10}-\frac{58987}{350784}a^{8}+\frac{20147}{18792}a^{6}+\frac{50927}{29232}a^{4}+\frac{358}{87}a^{2}+\frac{96952}{16443}$, $\frac{207983}{269402112}a^{30}+\frac{2111}{6414336}a^{28}+\frac{128351}{29933568}a^{26}-\frac{28549}{9621504}a^{24}+\frac{560369}{44900352}a^{22}-\frac{3192817}{89800704}a^{20}+\frac{2298271}{134701056}a^{18}-\frac{5994853}{29933568}a^{16}+\frac{855707}{29933568}a^{14}-\frac{42193895}{67350528}a^{12}+\frac{629273}{1403136}a^{10}-\frac{377921}{467712}a^{8}+\frac{249017}{75168}a^{6}+\frac{234599}{87696}a^{4}+\frac{36883}{3132}a^{2}+\frac{174445}{16443}$, $\frac{949}{44900352}a^{30}-\frac{883}{801792}a^{28}+\frac{10211}{14966784}a^{26}-\frac{21809}{3207168}a^{24}+\frac{84631}{7483392}a^{22}-\frac{1213733}{44900352}a^{20}+\frac{862471}{11225088}a^{18}-\frac{1368305}{14966784}a^{16}+\frac{5459333}{14966784}a^{14}-\frac{8483459}{22450176}a^{12}+\frac{6709453}{5612544}a^{10}-\frac{53147}{29232}a^{8}+\frac{127811}{50112}a^{6}-\frac{8657}{1218}a^{4}+\frac{3311}{1566}a^{2}-\frac{98684}{5481}$, $\frac{2321}{44900352}a^{31}+\frac{4201}{29933568}a^{30}+\frac{1231}{6414336}a^{29}+\frac{119}{2138112}a^{28}+\frac{9703}{44900352}a^{27}+\frac{64147}{89800704}a^{26}+\frac{4981}{6414336}a^{25}-\frac{191}{356352}a^{24}-\frac{19205}{22450176}a^{23}+\frac{75919}{44900352}a^{22}+\frac{69353}{44900352}a^{21}-\frac{508799}{89800704}a^{20}-\frac{142619}{14966784}a^{19}+\frac{63775}{44900352}a^{18}-\frac{24991}{14966784}a^{17}-\frac{856715}{29933568}a^{16}-\frac{46999}{935424}a^{15}+\frac{2213}{3325952}a^{14}+\frac{513895}{44900352}a^{13}-\frac{184909}{2494464}a^{12}-\frac{1565329}{11225088}a^{11}+\frac{31609}{467712}a^{10}+\frac{544529}{2806272}a^{9}-\frac{11507}{701568}a^{8}-\frac{8933}{50112}a^{7}+\frac{1697}{4176}a^{6}+\frac{99257}{87696}a^{5}+\frac{14689}{21924}a^{4}+\frac{2245}{6264}a^{3}+\frac{2563}{1566}a^{2}+\frac{3985}{1218}a+\frac{13306}{5481}$, $\frac{431}{3207168}a^{31}-\frac{15815}{29933568}a^{30}+\frac{367}{712704}a^{29}-\frac{703}{2138112}a^{28}+\frac{923}{1603584}a^{27}-\frac{275749}{89800704}a^{26}+\frac{13819}{6414336}a^{25}+\frac{89}{66816}a^{24}-\frac{139}{66816}a^{23}-\frac{367249}{44900352}a^{22}+\frac{3437}{801792}a^{21}+\frac{2026361}{89800704}a^{20}-\frac{178451}{6414336}a^{19}-\frac{261019}{44900352}a^{18}-\frac{95}{59392}a^{17}+\frac{4009997}{29933568}a^{16}-\frac{299081}{2138112}a^{15}+\frac{129683}{9977856}a^{14}+\frac{184871}{6414336}a^{13}+\frac{780667}{1870848}a^{12}-\frac{69059}{178176}a^{11}-\frac{66895}{311808}a^{10}+\frac{221215}{400896}a^{9}+\frac{639103}{1403136}a^{8}-\frac{40981}{100224}a^{7}-\frac{17093}{8352}a^{6}+\frac{13531}{4176}a^{5}-\frac{54613}{21924}a^{4}+\frac{7361}{6264}a^{3}-\frac{26393}{3132}a^{2}+\frac{14129}{1566}a-\frac{51763}{5481}$, $\frac{2213}{6193152}a^{31}-\frac{71}{442368}a^{30}-\frac{113}{442368}a^{29}-\frac{275}{221184}a^{28}+\frac{14093}{6193152}a^{27}-\frac{127}{442368}a^{26}-\frac{791}{221184}a^{25}-\frac{247}{36864}a^{24}+\frac{9347}{1032192}a^{23}+\frac{1805}{221184}a^{22}-\frac{5923}{229376}a^{21}-\frac{8885}{442368}a^{20}+\frac{103093}{3096576}a^{19}+\frac{16873}{221184}a^{18}-\frac{253045}{2064384}a^{17}-\frac{897}{16384}a^{16}+\frac{278939}{2064384}a^{15}+\frac{18605}{49152}a^{14}-\frac{617657}{1548288}a^{13}-\frac{25981}{110592}a^{12}+\frac{232759}{387072}a^{11}+\frac{31891}{27648}a^{10}-\frac{92663}{96768}a^{9}-\frac{2873}{1728}a^{8}+\frac{509}{216}a^{7}+\frac{587}{288}a^{6}-\frac{41}{42}a^{5}-\frac{1703}{216}a^{4}+\frac{103}{18}a^{3}-\frac{25}{108}a^{2}-\frac{160}{189}a-\frac{566}{27}$, $\frac{4513}{9977856}a^{31}-\frac{3883}{7483392}a^{30}+\frac{265}{267264}a^{29}+\frac{5}{50112}a^{28}+\frac{192037}{89800704}a^{27}-\frac{71941}{22450176}a^{26}+\frac{20023}{6414336}a^{25}+\frac{6073}{1603584}a^{24}-\frac{42227}{44900352}a^{23}-\frac{130969}{11225088}a^{22}-\frac{86153}{89800704}a^{21}+\frac{672433}{22450176}a^{20}-\frac{362045}{7483392}a^{19}-\frac{177641}{5612544}a^{18}-\frac{1432829}{29933568}a^{17}+\frac{378703}{2494464}a^{16}-\frac{2564965}{9977856}a^{15}-\frac{273715}{2494464}a^{14}-\frac{1341077}{14966784}a^{13}+\frac{1900697}{3741696}a^{12}-\frac{1177433}{1870848}a^{11}-\frac{1695061}{2806272}a^{10}+\frac{2433211}{2806272}a^{9}+\frac{1469365}{1403136}a^{8}-\frac{8485}{100224}a^{7}-\frac{17221}{6264}a^{6}+\frac{1231373}{175392}a^{5}+\frac{6809}{87696}a^{4}+\frac{30697}{6264}a^{3}-\frac{12503}{1566}a^{2}+\frac{12260}{609}a-\frac{15268}{5481}$, $\frac{34625}{89800704}a^{31}+\frac{15}{59392}a^{30}+\frac{863}{2138112}a^{29}+\frac{761}{1069056}a^{28}+\frac{193297}{89800704}a^{27}+\frac{175}{133632}a^{26}-\frac{155}{1603584}a^{25}+\frac{10103}{3207168}a^{24}+\frac{60535}{14966784}a^{23}-\frac{13}{33408}a^{22}-\frac{1112773}{89800704}a^{21}+\frac{373}{89088}a^{20}-\frac{260381}{44900352}a^{19}-\frac{117671}{3207168}a^{18}-\frac{2380153}{29933568}a^{17}-\frac{2677}{133632}a^{16}-\frac{1740653}{29933568}a^{15}-\frac{209729}{1069056}a^{14}-\frac{2737417}{11225088}a^{13}-\frac{23239}{1069056}a^{12}-\frac{28691}{1870848}a^{11}-\frac{144007}{267264}a^{10}-\frac{52517}{2806272}a^{9}+\frac{19615}{33408}a^{8}+\frac{115247}{100224}a^{7}-\frac{5735}{12528}a^{6}+\frac{2265}{812}a^{5}+\frac{424}{87}a^{4}+\frac{30647}{6264}a^{3}+\frac{3163}{1044}a^{2}+\frac{46402}{5481}a+\frac{11558}{783}$, $\frac{39047}{538804224}a^{31}-\frac{114379}{269402112}a^{30}-\frac{6173}{12828672}a^{29}+\frac{1037}{2138112}a^{28}+\frac{114005}{179601408}a^{27}-\frac{250345}{89800704}a^{26}-\frac{67339}{19243008}a^{25}+\frac{13499}{2405376}a^{24}+\frac{59249}{9977856}a^{23}-\frac{578485}{44900352}a^{22}-\frac{112299}{6651904}a^{21}+\frac{3143789}{89800704}a^{20}+\frac{9851083}{269402112}a^{19}-\frac{7077101}{134701056}a^{18}-\frac{3688093}{59867136}a^{17}+\frac{538729}{3325952}a^{16}+\frac{10495339}{59867136}a^{15}-\frac{753035}{3325952}a^{14}-\frac{30240593}{134701056}a^{13}+\frac{4622585}{8418816}a^{12}+\frac{1781401}{2806272}a^{11}-\frac{455201}{467712}a^{10}-\frac{78643}{87696}a^{9}+\frac{2008483}{1403136}a^{8}+\frac{249143}{150336}a^{7}-\frac{498929}{150336}a^{6}-\frac{184763}{58464}a^{5}+\frac{28069}{10962}a^{4}+\frac{265}{116}a^{3}-\frac{22043}{3132}a^{2}-\frac{272249}{32886}a+\frac{71143}{16443}$, $\frac{4999}{269402112}a^{31}-\frac{8399}{269402112}a^{30}+\frac{161}{2138112}a^{29}+\frac{2165}{6414336}a^{28}+\frac{3823}{29933568}a^{27}-\frac{14447}{29933568}a^{26}+\frac{4657}{9621504}a^{25}+\frac{22579}{9621504}a^{24}-\frac{2539}{44900352}a^{23}-\frac{210817}{44900352}a^{22}+\frac{171799}{89800704}a^{21}+\frac{1002881}{89800704}a^{20}-\frac{716845}{134701056}a^{19}-\frac{3624211}{134701056}a^{18}+\frac{29915}{29933568}a^{17}+\frac{1273493}{29933568}a^{16}-\frac{275095}{9977856}a^{15}-\frac{3847955}{29933568}a^{14}+\frac{381761}{67350528}a^{13}+\frac{11967377}{67350528}a^{12}-\frac{137645}{1247232}a^{11}-\frac{1143473}{2806272}a^{10}+\frac{90107}{935424}a^{9}+\frac{326107}{467712}a^{8}-\frac{57257}{300672}a^{7}-\frac{165175}{150336}a^{6}+\frac{105649}{175392}a^{5}+\frac{106039}{43848}a^{4}+\frac{11}{3132}a^{3}-\frac{5615}{3132}a^{2}+\frac{38432}{16443}a+\frac{85325}{16443}$, $\frac{4999}{269402112}a^{31}+\frac{8399}{269402112}a^{30}+\frac{161}{2138112}a^{29}-\frac{2165}{6414336}a^{28}+\frac{3823}{29933568}a^{27}+\frac{14447}{29933568}a^{26}+\frac{4657}{9621504}a^{25}-\frac{22579}{9621504}a^{24}-\frac{2539}{44900352}a^{23}+\frac{210817}{44900352}a^{22}+\frac{171799}{89800704}a^{21}-\frac{1002881}{89800704}a^{20}-\frac{716845}{134701056}a^{19}+\frac{3624211}{134701056}a^{18}+\frac{29915}{29933568}a^{17}-\frac{1273493}{29933568}a^{16}-\frac{275095}{9977856}a^{15}+\frac{3847955}{29933568}a^{14}+\frac{381761}{67350528}a^{13}-\frac{11967377}{67350528}a^{12}-\frac{137645}{1247232}a^{11}+\frac{1143473}{2806272}a^{10}+\frac{90107}{935424}a^{9}-\frac{326107}{467712}a^{8}-\frac{57257}{300672}a^{7}+\frac{165175}{150336}a^{6}+\frac{105649}{175392}a^{5}-\frac{106039}{43848}a^{4}+\frac{11}{3132}a^{3}+\frac{5615}{3132}a^{2}+\frac{38432}{16443}a-\frac{85325}{16443}$ 3713/44900352a^31 + 259/801792a^29 + 16141/44900352a^27 + 491/356352a^25 - 27499/22450176a^23 + 123119/44900352a^21 - 205325/11225088a^19 + 1051/14966784a^17 - 1341583/14966784a^15 + 390101/22450176a^13 - 696347/2806272a^11 + 125497/350784a^9 - 7705/33408a^7 + 92447/43848a^5 + 1279/1566a^3 + 31519/5481a - 1, 50893/269402112a^30 + 1117/6414336a^28 + 37229/29933568a^26 - 1751/9621504a^24 + 108107/44900352a^22 - 69499/9977856a^20 - 90547/134701056a^18 - 1444543/29933568a^16 - 790927/29933568a^14 - 9107617/67350528a^12 + 71059/5612544a^10 - 16067/155904a^8 + 101327/150336a^6 + 118913/87696a^4 + 2735/1044a^2 + 84104/16443, 1241/7483392a^30 - 803/1069056a^28 + 32261/22450176a^26 - 5845/1069056a^24 + 117083/11225088a^22 - 622273/22450176a^20 + 1307767/22450176a^18 - 822421/7483392a^16 + 336703/1247232a^14 - 1014073/2494464a^12 + 299479/311808a^10 - 1984253/1403136a^8 + 22423/8352a^6 - 95407/21924a^4 + 13675/3132a^2 - 57856/5481, 41/442368a^30 - 229/221184a^28 + 505/442368a^26 - 137/18432a^24 + 2749/221184a^22 - 14605/442368a^20 + 17795/221184a^18 - 5899/49152a^16 + 18665/49152a^14 - 25303/55296a^12 + 36347/27648a^10 - 13699/6912a^8 + 233/72a^6 - 2993/432a^4 + 421/108a^2 - 469/27, 87389/269402112a^30 + 1505/6414336a^28 + 154087/89800704a^26 - 6169/9621504a^24 + 67609/14966784a^22 - 36713/3325952a^20 + 48745/134701056a^18 - 691573/9977856a^16 - 530263/29933568a^14 - 14639735/67350528a^12 + 347105/5612544a^10 - 58987/350784a^8 + 20147/18792a^6 + 50927/29232a^4 + 358/87a^2 + 96952/16443, 207983/269402112a^30 + 2111/6414336a^28 + 128351/29933568a^26 - 28549/9621504a^24 + 560369/44900352a^22 - 3192817/89800704a^20 + 2298271/134701056a^18 - 5994853/29933568a^16 + 855707/29933568a^14 - 42193895/67350528a^12 + 629273/1403136a^10 - 377921/467712a^8 + 249017/75168a^6 + 234599/87696a^4 + 36883/3132a^2 + 174445/16443, 949/44900352a^30 - 883/801792a^28 + 10211/14966784a^26 - 21809/3207168a^24 + 84631/7483392a^22 - 1213733/44900352a^20 + 862471/11225088a^18 - 1368305/14966784a^16 + 5459333/14966784a^14 - 8483459/22450176a^12 + 6709453/5612544a^10 - 53147/29232a^8 + 127811/50112a^6 - 8657/1218a^4 + 3311/1566a^2 - 98684/5481, 2321/44900352a^31 + 4201/29933568a^30 + 1231/6414336a^29 + 119/2138112a^28 + 9703/44900352a^27 + 64147/89800704a^26 + 4981/6414336a^25 - 191/356352a^24 - 19205/22450176a^23 + 75919/44900352a^22 + 69353/44900352a^21 - 508799/89800704a^20 - 142619/14966784a^19 + 63775/44900352a^18 - 24991/14966784a^17 - 856715/29933568a^16 - 46999/935424a^15 + 2213/3325952a^14 + 513895/44900352a^13 - 184909/2494464a^12 - 1565329/11225088a^11 + 31609/467712a^10 + 544529/2806272a^9 - 11507/701568a^8 - 8933/50112a^7 + 1697/4176a^6 + 99257/87696a^5 + 14689/21924a^4 + 2245/6264a^3 + 2563/1566a^2 + 3985/1218a + 13306/5481, 431/3207168a^31 - 15815/29933568a^30 + 367/712704a^29 - 703/2138112a^28 + 923/1603584a^27 - 275749/89800704a^26 + 13819/6414336a^25 + 89/66816a^24 - 139/66816a^23 - 367249/44900352a^22 + 3437/801792a^21 + 2026361/89800704a^20 - 178451/6414336a^19 - 261019/44900352a^18 - 95/59392a^17 + 4009997/29933568a^16 - 299081/2138112a^15 + 129683/9977856a^14 + 184871/6414336a^13 + 780667/1870848a^12 - 69059/178176a^11 - 66895/311808a^10 + 221215/400896a^9 + 639103/1403136a^8 - 40981/100224a^7 - 17093/8352a^6 + 13531/4176a^5 - 54613/21924a^4 + 7361/6264a^3 - 26393/3132a^2 + 14129/1566a - 51763/5481, 2213/6193152a^31 - 71/442368a^30 - 113/442368a^29 - 275/221184a^28 + 14093/6193152a^27 - 127/442368a^26 - 791/221184a^25 - 247/36864a^24 + 9347/1032192a^23 + 1805/221184a^22 - 5923/229376a^21 - 8885/442368a^20 + 103093/3096576a^19 + 16873/221184a^18 - 253045/2064384a^17 - 897/16384a^16 + 278939/2064384a^15 + 18605/49152a^14 - 617657/1548288a^13 - 25981/110592a^12 + 232759/387072a^11 + 31891/27648a^10 - 92663/96768a^9 - 2873/1728a^8 + 509/216a^7 + 587/288a^6 - 41/42a^5 - 1703/216a^4 + 103/18a^3 - 25/108a^2 - 160/189a - 566/27, 4513/9977856a^31 - 3883/7483392a^30 + 265/267264a^29 + 5/50112a^28 + 192037/89800704a^27 - 71941/22450176a^26 + 20023/6414336a^25 + 6073/1603584a^24 - 42227/44900352a^23 - 130969/11225088a^22 - 86153/89800704a^21 + 672433/22450176a^20 - 362045/7483392a^19 - 177641/5612544a^18 - 1432829/29933568a^17 + 378703/2494464a^16 - 2564965/9977856a^15 - 273715/2494464a^14 - 1341077/14966784a^13 + 1900697/3741696a^12 - 1177433/1870848a^11 - 1695061/2806272a^10 + 2433211/2806272a^9 + 1469365/1403136a^8 - 8485/100224a^7 - 17221/6264a^6 + 1231373/175392a^5 + 6809/87696a^4 + 30697/6264a^3 - 12503/1566a^2 + 12260/609a - 15268/5481, 34625/89800704a^31 + 15/59392a^30 + 863/2138112a^29 + 761/1069056a^28 + 193297/89800704a^27 + 175/133632a^26 - 155/1603584a^25 + 10103/3207168a^24 + 60535/14966784a^23 - 13/33408a^22 - 1112773/89800704a^21 + 373/89088a^20 - 260381/44900352a^19 - 117671/3207168a^18 - 2380153/29933568a^17 - 2677/133632a^16 - 1740653/29933568a^15 - 209729/1069056a^14 - 2737417/11225088a^13 - 23239/1069056a^12 - 28691/1870848a^11 - 144007/267264a^10 - 52517/2806272a^9 + 19615/33408a^8 + 115247/100224a^7 - 5735/12528a^6 + 2265/812a^5 + 424/87a^4 + 30647/6264a^3 + 3163/1044a^2 + 46402/5481a + 11558/783, 39047/538804224a^31 - 114379/269402112a^30 - 6173/12828672a^29 + 1037/2138112a^28 + 114005/179601408a^27 - 250345/89800704a^26 - 67339/19243008a^25 + 13499/2405376a^24 + 59249/9977856a^23 - 578485/44900352a^22 - 112299/6651904a^21 + 3143789/89800704a^20 + 9851083/269402112a^19 - 7077101/134701056a^18 - 3688093/59867136a^17 + 538729/3325952a^16 + 10495339/59867136a^15 - 753035/3325952a^14 - 30240593/134701056a^13 + 4622585/8418816a^12 + 1781401/2806272a^11 - 455201/467712a^10 - 78643/87696a^9 + 2008483/1403136a^8 + 249143/150336a^7 - 498929/150336a^6 - 184763/58464a^5 + 28069/10962a^4 + 265/116a^3 - 22043/3132a^2 - 272249/32886a + 71143/16443, 4999/269402112a^31 - 8399/269402112a^30 + 161/2138112a^29 + 2165/6414336a^28 + 3823/29933568a^27 - 14447/29933568a^26 + 4657/9621504a^25 + 22579/9621504a^24 - 2539/44900352a^23 - 210817/44900352a^22 + 171799/89800704a^21 + 1002881/89800704a^20 - 716845/134701056a^19 - 3624211/134701056a^18 + 29915/29933568a^17 + 1273493/29933568a^16 - 275095/9977856a^15 - 3847955/29933568a^14 + 381761/67350528a^13 + 11967377/67350528a^12 - 137645/1247232a^11 - 1143473/2806272a^10 + 90107/935424a^9 + 326107/467712a^8 - 57257/300672a^7 - 165175/150336a^6 + 105649/175392a^5 + 106039/43848a^4 + 11/3132a^3 - 5615/3132a^2 + 38432/16443a + 85325/16443, 4999/269402112a^31 + 8399/269402112a^30 + 161/2138112a^29 - 2165/6414336a^28 + 3823/29933568a^27 + 14447/29933568a^26 + 4657/9621504a^25 - 22579/9621504a^24 - 2539/44900352a^23 + 210817/44900352a^22 + 171799/89800704a^21 - 1002881/89800704a^20 - 716845/134701056a^19 + 3624211/134701056a^18 + 29915/29933568a^17 - 1273493/29933568a^16 - 275095/9977856a^15 + 3847955/29933568a^14 + 381761/67350528a^13 - 11967377/67350528a^12 - 137645/1247232a^11 + 1143473/2806272a^10 + 90107/935424a^9 - 326107/467712a^8 - 57257/300672a^7 + 165175/150336a^6 + 105649/175392a^5 - 106039/43848a^4 + 11/3132a^3 + 5615/3132a^2 + 38432/16443a - 85325/16443 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 333896890255.2903 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 333896890255.2903 \cdot 20}{12\cdot\sqrt{44580746322432044939368312527587514829553937678336}}\cr\approx \mathstrut & 0.491773758057361 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^32 + 2*x^30 + 9*x^28 + 4*x^26 + 26*x^24 - 45*x^22 + 10*x^20 - 403*x^18 - 171*x^16 - 1612*x^14 + 160*x^12 - 2880*x^10 + 6656*x^8 + 4096*x^6 + 36864*x^4 + 32768*x^2 + 65536)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^32 + 2*x^30 + 9*x^28 + 4*x^26 + 26*x^24 - 45*x^22 + 10*x^20 - 403*x^18 - 171*x^16 - 1612*x^14 + 160*x^12 - 2880*x^10 + 6656*x^8 + 4096*x^6 + 36864*x^4 + 32768*x^2 + 65536, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 + 2*x^30 + 9*x^28 + 4*x^26 + 26*x^24 - 45*x^22 + 10*x^20 - 403*x^18 - 171*x^16 - 1612*x^14 + 160*x^12 - 2880*x^10 + 6656*x^8 + 4096*x^6 + 36864*x^4 + 32768*x^2 + 65536);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 2*x^30 + 9*x^28 + 4*x^26 + 26*x^24 - 45*x^22 + 10*x^20 - 403*x^18 - 171*x^16 - 1612*x^14 + 160*x^12 - 2880*x^10 + 6656*x^8 + 4096*x^6 + 36864*x^4 + 32768*x^2 + 65536);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_2^2\times \SL(2,3)$ (as 32T405):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 96

The 28 conjugacy class representatives for $C_2^2\times \SL(2,3)$

Character table for $C_2^2\times \SL(2,3)$

Intermediate fields

$\Q(\sqrt{3}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{-1}) $, 4.4.33489.1, $\Q(\zeta_{12})$, 8.0.2583966230784.1, 8.0.287107358976.16, 8.8.10093618089.1, 8.8.287107358976.1, 8.0.10093618089.1, 8.0.1121513121.1, 8.8.2583966230784.1, 16.0.6676881481832071949254656.4, 16.0.6676881481832071949254656.1, 16.0.6676881481832071949254656.5, 16.0.82430635578173727768576.3, 16.0.6676881481832071949254656.3, 16.16.6676881481832071949254656.1, 16.0.101881126126588011921.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	${\href{/padicField/5.6.0.1}{6} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$	${\href{/padicField/7.6.0.1}{6} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$	${\href{/padicField/11.4.0.1}{4} }^{8}$	${\href{/padicField/13.6.0.1}{6} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$	${\href{/padicField/17.6.0.1}{6} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$	${\href{/padicField/19.6.0.1}{6} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$	${\href{/padicField/23.4.0.1}{4} }^{8}$	${\href{/padicField/29.6.0.1}{6} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$	${\href{/padicField/31.6.0.1}{6} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$	${\href{/padicField/37.4.0.1}{4} }^{8}$	${\href{/padicField/41.4.0.1}{4} }^{8}$	${\href{/padicField/43.6.0.1}{6} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$	${\href{/padicField/47.6.0.1}{6} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$	${\href{/padicField/53.4.0.1}{4} }^{8}$	${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.4.4.1	$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
	2.4.4.1	$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
	2.12.12.26	$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$	$2$	$6$	$12$	$C_6\times C_2$	$[2]^{6}$
	2.12.12.26	$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$	$2$	$6$	$12$	$C_6\times C_2$	$[2]^{6}$
$3$ 3	3.8.6.1	$x^{8} + 9$	$4$	$2$	$6$	$Q_8$	$[\ ]_{4}^{2}$
	3.8.6.1	$x^{8} + 9$	$4$	$2$	$6$	$Q_8$	$[\ ]_{4}^{2}$
	3.8.6.1	$x^{8} + 9$	$4$	$2$	$6$	$Q_8$	$[\ ]_{4}^{2}$
	3.8.6.1	$x^{8} + 9$	$4$	$2$	$6$	$Q_8$	$[\ ]_{4}^{2}$
$61$ 61	61.2.0.1	$x^{2} + 60 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	61.2.0.1	$x^{2} + 60 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	61.2.0.1	$x^{2} + 60 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	61.2.0.1	$x^{2} + 60 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	61.6.4.1	$x^{6} + 180 x^{5} + 10806 x^{4} + 216842 x^{3} + 32592 x^{2} + 658788 x + 13157769$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
	61.6.4.1	$x^{6} + 180 x^{5} + 10806 x^{4} + 216842 x^{3} + 32592 x^{2} + 658788 x + 13157769$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
	61.6.4.1	$x^{6} + 180 x^{5} + 10806 x^{4} + 216842 x^{3} + 32592 x^{2} + 658788 x + 13157769$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
	61.6.4.1	$x^{6} + 180 x^{5} + 10806 x^{4} + 216842 x^{3} + 32592 x^{2} + 658788 x + 13157769$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$

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