LMFDB - Number field 32.0.11242761639679742219392447654740052674625150753177600000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 11*x^28 + 62*x^24 - 260*x^20 + 1041*x^16 - 4160*x^12 + 15872*x^8 - 45056*x^4 + 65536)

gp: K = bnfinit(y^32 - 11*y^28 + 62*y^24 - 260*y^20 + 1041*y^16 - 4160*y^12 + 15872*y^8 - 45056*y^4 + 65536, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 11*x^28 + 62*x^24 - 260*x^20 + 1041*x^16 - 4160*x^12 + 15872*x^8 - 45056*x^4 + 65536);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 11*x^28 + 62*x^24 - 260*x^20 + 1041*x^16 - 4160*x^12 + 15872*x^8 - 45056*x^4 + 65536)

$ x^{32} - 11x^{28} + 62x^{24} - 260x^{20} + 1041x^{16} - 4160x^{12} + 15872x^{8} - 45056x^{4} + 65536 $ x^32 - 11*x^28 + 62*x^24 - 260*x^20 + 1041*x^16 - 4160*x^12 + 15872*x^8 - 45056*x^4 + 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:	$11242761639679742219392447654740052674625150753177600000000$ 11242761639679742219392447654740052674625150753177600000000 $\medspace = 2^{64}\cdot 5^{8}\cdot 29^{16}\cdot 281^{4}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$65.18$	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:	$2^{2}5^{1/2}29^{1/2}281^{1/2}\approx 807.4156302673364$
Ramified primes:	$2$, $5$, $29$, $281$ 2, 5, 29, 281	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{12}-\frac{1}{5}a^{8}+\frac{2}{5}a^{4}+\frac{1}{5}$, $\frac{1}{10}a^{17}-\frac{3}{10}a^{13}+\frac{2}{5}a^{9}+\frac{1}{5}a^{5}+\frac{1}{10}a$, $\frac{1}{20}a^{18}-\frac{3}{20}a^{14}-\frac{3}{10}a^{10}-\frac{2}{5}a^{6}+\frac{1}{20}a^{2}$, $\frac{1}{40}a^{19}-\frac{3}{40}a^{15}+\frac{7}{20}a^{11}+\frac{3}{10}a^{7}+\frac{1}{40}a^{3}$, $\frac{1}{80}a^{20}+\frac{1}{16}a^{16}+\frac{3}{8}a^{12}-\frac{9}{20}a^{8}+\frac{17}{80}a^{4}-\frac{2}{5}$, $\frac{1}{160}a^{21}+\frac{1}{32}a^{17}-\frac{5}{16}a^{13}-\frac{9}{40}a^{9}+\frac{17}{160}a^{5}+\frac{3}{10}a$, $\frac{1}{320}a^{22}+\frac{1}{64}a^{18}-\frac{5}{32}a^{14}-\frac{9}{80}a^{10}+\frac{17}{320}a^{6}-\frac{7}{20}a^{2}$, $\frac{1}{640}a^{23}+\frac{1}{128}a^{19}+\frac{27}{64}a^{15}-\frac{9}{160}a^{11}-\frac{303}{640}a^{7}+\frac{13}{40}a^{3}$, $\frac{1}{24320}a^{24}-\frac{59}{24320}a^{20}+\frac{679}{12160}a^{16}+\frac{43}{1216}a^{12}-\frac{10287}{24320}a^{8}-\frac{279}{1520}a^{4}+\frac{16}{95}$, $\frac{1}{48640}a^{25}-\frac{59}{48640}a^{21}+\frac{679}{24320}a^{17}-\frac{1173}{2432}a^{13}+\frac{14033}{48640}a^{9}-\frac{279}{3040}a^{5}-\frac{79}{190}a$, $\frac{1}{97280}a^{26}-\frac{59}{97280}a^{22}+\frac{679}{48640}a^{18}+\frac{1259}{4864}a^{14}+\frac{14033}{97280}a^{10}+\frac{2761}{6080}a^{6}-\frac{79}{380}a^{2}$, $\frac{1}{194560}a^{27}-\frac{59}{194560}a^{23}+\frac{679}{97280}a^{19}+\frac{1259}{9728}a^{15}+\frac{14033}{194560}a^{11}-\frac{3319}{12160}a^{7}+\frac{301}{760}a^{3}$, $\frac{1}{13619200}a^{28}+\frac{7}{389120}a^{24}+\frac{40351}{6809600}a^{20}+\frac{133823}{3404800}a^{16}+\frac{3388433}{13619200}a^{12}-\frac{80093}{212800}a^{8}+\frac{43}{380}a^{4}+\frac{16}{175}$, $\frac{1}{27238400}a^{29}+\frac{7}{778240}a^{25}+\frac{40351}{13619200}a^{21}+\frac{133823}{6809600}a^{17}-\frac{10230767}{27238400}a^{13}-\frac{80093}{425600}a^{9}+\frac{43}{760}a^{5}+\frac{8}{175}a$, $\frac{1}{54476800}a^{30}+\frac{7}{1556480}a^{26}+\frac{40351}{27238400}a^{22}+\frac{133823}{13619200}a^{18}-\frac{10230767}{54476800}a^{14}-\frac{80093}{851200}a^{10}-\frac{717}{1520}a^{6}+\frac{4}{175}a^{2}$, $\frac{1}{108953600}a^{31}+\frac{7}{3112960}a^{27}+\frac{40351}{54476800}a^{23}+\frac{133823}{27238400}a^{19}+\frac{44246033}{108953600}a^{15}+\frac{771107}{1702400}a^{11}-\frac{717}{3040}a^{7}-\frac{171}{350}a^{3}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15, 1/5*a^16 + 2/5*a^12 - 1/5*a^8 + 2/5*a^4 + 1/5, 1/10*a^17 - 3/10*a^13 + 2/5*a^9 + 1/5*a^5 + 1/10*a, 1/20*a^18 - 3/20*a^14 - 3/10*a^10 - 2/5*a^6 + 1/20*a^2, 1/40*a^19 - 3/40*a^15 + 7/20*a^11 + 3/10*a^7 + 1/40*a^3, 1/80*a^20 + 1/16*a^16 + 3/8*a^12 - 9/20*a^8 + 17/80*a^4 - 2/5, 1/160*a^21 + 1/32*a^17 - 5/16*a^13 - 9/40*a^9 + 17/160*a^5 + 3/10*a, 1/320*a^22 + 1/64*a^18 - 5/32*a^14 - 9/80*a^10 + 17/320*a^6 - 7/20*a^2, 1/640*a^23 + 1/128*a^19 + 27/64*a^15 - 9/160*a^11 - 303/640*a^7 + 13/40*a^3, 1/24320*a^24 - 59/24320*a^20 + 679/12160*a^16 + 43/1216*a^12 - 10287/24320*a^8 - 279/1520*a^4 + 16/95, 1/48640*a^25 - 59/48640*a^21 + 679/24320*a^17 - 1173/2432*a^13 + 14033/48640*a^9 - 279/3040*a^5 - 79/190*a, 1/97280*a^26 - 59/97280*a^22 + 679/48640*a^18 + 1259/4864*a^14 + 14033/97280*a^10 + 2761/6080*a^6 - 79/380*a^2, 1/194560*a^27 - 59/194560*a^23 + 679/97280*a^19 + 1259/9728*a^15 + 14033/194560*a^11 - 3319/12160*a^7 + 301/760*a^3, 1/13619200*a^28 + 7/389120*a^24 + 40351/6809600*a^20 + 133823/3404800*a^16 + 3388433/13619200*a^12 - 80093/212800*a^8 + 43/380*a^4 + 16/175, 1/27238400*a^29 + 7/778240*a^25 + 40351/13619200*a^21 + 133823/6809600*a^17 - 10230767/27238400*a^13 - 80093/425600*a^9 + 43/760*a^5 + 8/175*a, 1/54476800*a^30 + 7/1556480*a^26 + 40351/27238400*a^22 + 133823/13619200*a^18 - 10230767/54476800*a^14 - 80093/851200*a^10 - 717/1520*a^6 + 4/175*a^2, 1/108953600*a^31 + 7/3112960*a^27 + 40351/54476800*a^23 + 133823/27238400*a^19 + 44246033/108953600*a^15 + 771107/1702400*a^11 - 717/3040*a^7 - 171/350*a^3

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$5$

Class group and class number

$C_{3}\times C_{186}$, which has order $558$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Relative class number: $558$ (assuming GRH)

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{3351}{27238400} a^{29} + \frac{623}{778240} a^{25} - \frac{53201}{13619200} a^{21} + \frac{98687}{6809600} a^{17} - \frac{1407303}{27238400} a^{13} + \frac{379807}{1702400} a^{9} - \frac{2339}{3040} a^{5} + \frac{3818}{3325} a $ -(3351)/(27238400)a^(29) + (623)/(778240)a^(25) - (53201)/(13619200)a^(21) + (98687)/(6809600)a^(17) - (1407303)/(27238400)a^(13) + (379807)/(1702400)a^(9) - (2339)/(3040)a^(5) + (3818)/(3325)a (order $8$)	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:	$\frac{431}{27238400}a^{29}-\frac{279}{778240}a^{25}+\frac{25121}{13619200}a^{21}-\frac{43167}{6809600}a^{17}+\frac{646943}{27238400}a^{13}-\frac{35531}{851200}a^{9}+\frac{253}{1520}a^{5}-\frac{6281}{6650}a$, $\frac{543}{13619200}a^{31}-\frac{123}{389120}a^{27}+\frac{11163}{6809600}a^{23}-\frac{23721}{3404800}a^{19}+\frac{297279}{13619200}a^{15}-\frac{312479}{3404800}a^{11}+\frac{5777}{12160}a^{7}-\frac{18209}{26600}a^{3}$, $\frac{723}{10895360}a^{30}-\frac{3351}{27238400}a^{29}-\frac{743}{1556480}a^{26}+\frac{623}{778240}a^{25}+\frac{10989}{5447680}a^{22}-\frac{53201}{13619200}a^{21}-\frac{22707}{2723840}a^{18}+\frac{98687}{6809600}a^{17}+\frac{440067}{10895360}a^{14}-\frac{1407303}{27238400}a^{13}-\frac{2833}{21280}a^{10}+\frac{379807}{1702400}a^{9}+\frac{2929}{6080}a^{6}-\frac{2339}{3040}a^{5}-\frac{2311}{2660}a^{2}+\frac{3818}{3325}a-1$, $\frac{1}{77824}a^{28}+\frac{73}{389120}a^{24}-\frac{33}{38912}a^{20}-\frac{453}{97280}a^{16}+\frac{7253}{389120}a^{12}+\frac{237}{6080}a^{8}-\frac{59}{304}a^{4}-\frac{12}{95}$, $\frac{3443}{27238400}a^{31}+\frac{559}{2723840}a^{30}-\frac{4943}{13619200}a^{29}-\frac{863}{778240}a^{27}-\frac{491}{389120}a^{26}+\frac{9}{4096}a^{25}+\frac{60463}{13619200}a^{23}+\frac{8397}{1361920}a^{22}-\frac{68113}{6809600}a^{21}-\frac{131821}{6809600}a^{19}-\frac{15691}{680960}a^{18}+\frac{133871}{3404800}a^{17}+\frac{1978579}{27238400}a^{15}+\frac{59251}{544768}a^{14}-\frac{2183999}{13619200}a^{13}-\frac{2180579}{6809600}a^{11}-\frac{127317}{340480}a^{10}+\frac{142729}{212800}a^{9}+\frac{181}{160}a^{7}+\frac{8589}{6080}a^{6}-\frac{7101}{3040}a^{5}-\frac{68367}{26600}a^{3}-\frac{5647}{2660}a^{2}+\frac{13598}{3325}a$, $\frac{321}{3404800}a^{31}-\frac{1}{28672}a^{30}-\frac{39}{194560}a^{28}-\frac{61}{97280}a^{27}+\frac{13}{20480}a^{26}+\frac{221}{194560}a^{24}+\frac{3441}{1702400}a^{23}-\frac{45}{14336}a^{22}-\frac{77}{19456}a^{20}-\frac{7347}{851200}a^{19}+\frac{87}{7168}a^{18}+\frac{1071}{48640}a^{16}+\frac{138433}{3404800}a^{15}-\frac{5861}{143360}a^{14}-\frac{15959}{194560}a^{12}-\frac{111043}{851200}a^{11}+\frac{6633}{35840}a^{10}+\frac{659}{2432}a^{8}+\frac{1211}{3040}a^{7}-\frac{33}{40}a^{6}-\frac{1649}{1520}a^{4}-\frac{8237}{26600}a^{3}+\frac{51}{28}a^{2}+\frac{42}{19}$, $\frac{599}{108953600}a^{31}+\frac{159}{3404800}a^{29}+\frac{3}{194560}a^{28}+\frac{113}{3112960}a^{27}-\frac{3}{5120}a^{25}-\frac{169}{194560}a^{24}-\frac{47231}{54476800}a^{23}+\frac{2797}{851200}a^{21}+\frac{117}{19456}a^{20}+\frac{99857}{27238400}a^{19}-\frac{2207}{212800}a^{17}-\frac{179}{9728}a^{16}-\frac{505593}{108953600}a^{15}+\frac{144167}{3404800}a^{13}+\frac{14931}{194560}a^{12}+\frac{94887}{6809600}a^{11}-\frac{429899}{1702400}a^{9}-\frac{9529}{24320}a^{8}-\frac{19}{128}a^{7}+\frac{2987}{3040}a^{5}+\frac{377}{380}a^{4}+\frac{6711}{26600}a^{3}-\frac{5701}{3325}a-\frac{174}{95}$, $\frac{6367}{108953600}a^{31}+\frac{43}{851200}a^{30}+\frac{67}{425600}a^{28}-\frac{1511}{3112960}a^{27}-\frac{3}{6080}a^{26}-\frac{21}{12160}a^{24}+\frac{103297}{54476800}a^{23}+\frac{1511}{851200}a^{22}+\frac{39}{5600}a^{20}-\frac{224159}{27238400}a^{19}-\frac{3457}{425600}a^{18}-\frac{383}{13300}a^{16}+\frac{3490511}{108953600}a^{15}+\frac{1061}{44800}a^{14}+\frac{50571}{425600}a^{12}-\frac{211571}{1702400}a^{11}-\frac{98251}{851200}a^{10}-\frac{112047}{212800}a^{8}+\frac{1219}{3040}a^{7}+\frac{1947}{6080}a^{6}+\frac{1381}{760}a^{4}-\frac{10851}{13300}a^{3}-\frac{10147}{13300}a^{2}-\frac{16059}{3325}$, $\frac{319}{54476800}a^{30}-\frac{103}{851200}a^{28}-\frac{967}{1556480}a^{26}-\frac{7}{24320}a^{24}+\frac{88289}{27238400}a^{22}+\frac{521}{212800}a^{20}-\frac{143743}{13619200}a^{18}-\frac{59}{13300}a^{16}+\frac{157173}{2867200}a^{14}+\frac{24561}{851200}a^{12}-\frac{173897}{851200}a^{10}-\frac{60577}{425600}a^{8}+\frac{5237}{6080}a^{6}+\frac{577}{760}a^{4}-\frac{31649}{13300}a^{2}-\frac{13372}{3325}$, $\frac{6367}{108953600}a^{31}+\frac{543}{2867200}a^{30}-\frac{3351}{13619200}a^{28}-\frac{1511}{3112960}a^{27}-\frac{1989}{1556480}a^{26}+\frac{623}{389120}a^{24}+\frac{103297}{54476800}a^{23}+\frac{161347}{27238400}a^{22}-\frac{53201}{6809600}a^{20}-\frac{224159}{27238400}a^{19}-\frac{310909}{13619200}a^{18}+\frac{98687}{3404800}a^{16}+\frac{3490511}{108953600}a^{15}+\frac{5014941}{54476800}a^{14}-\frac{1407303}{13619200}a^{12}-\frac{211571}{1702400}a^{11}-\frac{606447}{1702400}a^{10}+\frac{379807}{851200}a^{8}+\frac{1219}{3040}a^{7}+\frac{7607}{6080}a^{6}-\frac{2339}{1520}a^{4}-\frac{10851}{13300}a^{3}-\frac{26827}{13300}a^{2}+\frac{10961}{3325}$, $\frac{6911}{108953600}a^{31}-\frac{887}{6809600}a^{29}-\frac{519}{3112960}a^{27}+\frac{219}{194560}a^{25}+\frac{21601}{54476800}a^{23}-\frac{17207}{3404800}a^{21}-\frac{83327}{27238400}a^{19}+\frac{37549}{1702400}a^{17}+\frac{1414383}{108953600}a^{15}-\frac{458551}{6809600}a^{13}-\frac{52743}{1702400}a^{11}+\frac{508981}{1702400}a^{9}+\frac{621}{12160}a^{7}-\frac{1931}{1520}a^{5}+\frac{8539}{26600}a^{3}+\frac{7089}{3325}a$, $\frac{13753}{108953600}a^{31}+\frac{2329}{10895360}a^{30}-\frac{349}{1089536}a^{29}-\frac{2437}{6809600}a^{28}-\frac{2961}{3112960}a^{27}-\frac{2117}{1556480}a^{26}+\frac{1773}{778240}a^{25}+\frac{557}{194560}a^{24}+\frac{192823}{54476800}a^{23}+\frac{5939}{1089536}a^{22}-\frac{24079}{2723840}a^{21}-\frac{37187}{3404800}a^{20}-\frac{421961}{27238400}a^{19}-\frac{57457}{2723840}a^{18}+\frac{46353}{1361920}a^{17}+\frac{86509}{1702400}a^{16}+\frac{5897929}{108953600}a^{15}+\frac{937449}{10895360}a^{14}-\frac{785377}{5447680}a^{13}-\frac{1367701}{6809600}a^{12}-\frac{726513}{3404800}a^{11}-\frac{257937}{680960}a^{10}+\frac{49047}{85120}a^{9}+\frac{339419}{425600}a^{8}+\frac{9639}{12160}a^{7}+\frac{865}{608}a^{6}-\frac{6817}{3040}a^{5}-\frac{4411}{1520}a^{4}-\frac{12519}{13300}a^{3}-\frac{2677}{1330}a^{2}+\frac{5083}{1330}a+\frac{13434}{3325}$, $\frac{503}{5447680}a^{31}+\frac{1901}{7782400}a^{30}-\frac{543}{2723840}a^{29}-\frac{3603}{6809600}a^{28}-\frac{239}{778240}a^{27}-\frac{3299}{1556480}a^{26}+\frac{79}{77824}a^{25}+\frac{899}{194560}a^{24}+\frac{607}{544768}a^{23}+\frac{37331}{3891200}a^{22}-\frac{1233}{272384}a^{21}-\frac{70673}{3404800}a^{20}-\frac{733}{272384}a^{19}-\frac{79117}{1945600}a^{18}+\frac{2535}{136192}a^{17}+\frac{149031}{1702400}a^{16}+\frac{1515}{1089536}a^{15}+\frac{1307453}{7782400}a^{14}-\frac{166127}{2723840}a^{13}-\frac{137521}{358400}a^{12}+\frac{2913}{272384}a^{11}-\frac{9661}{15200}a^{10}+\frac{65943}{340480}a^{9}+\frac{1298187}{851200}a^{8}-\frac{651}{12160}a^{7}+\frac{2739}{1216}a^{6}-\frac{89}{160}a^{5}-\frac{8383}{1520}a^{4}+\frac{569}{665}a^{3}-\frac{2019}{475}a^{2}-\frac{278}{665}a+\frac{36111}{3325}$, $\frac{647}{3891200}a^{31}-\frac{1893}{10895360}a^{30}-\frac{2131}{5447680}a^{29}+\frac{319}{680960}a^{28}-\frac{1109}{778240}a^{27}+\frac{2209}{1556480}a^{26}+\frac{571}{155648}a^{25}-\frac{21}{5120}a^{24}+\frac{11827}{1945600}a^{23}-\frac{39699}{5447680}a^{22}-\frac{39629}{2723840}a^{21}+\frac{6431}{340480}a^{20}-\frac{25609}{972800}a^{19}+\frac{73117}{2723840}a^{18}+\frac{84051}{1361920}a^{17}-\frac{2353}{34048}a^{16}+\frac{367591}{3891200}a^{15}-\frac{1190389}{10895360}a^{14}-\frac{1345411}{5447680}a^{13}+\frac{205439}{680960}a^{12}-\frac{400791}{972800}a^{11}+\frac{58001}{136192}a^{10}+\frac{87133}{85120}a^{9}-\frac{42025}{34048}a^{8}+\frac{19533}{12160}a^{7}-\frac{2669}{1520}a^{6}-\frac{65}{19}a^{5}+\frac{6381}{1520}a^{4}-\frac{1546}{475}a^{3}+\frac{10841}{2660}a^{2}+\frac{4717}{665}a-\frac{1249}{133}$, $\frac{4329}{108953600}a^{31}-\frac{53}{7782400}a^{30}+\frac{183}{1361920}a^{29}+\frac{483}{1945600}a^{28}-\frac{1633}{3112960}a^{27}+\frac{123}{1556480}a^{26}-\frac{203}{194560}a^{25}-\frac{717}{389120}a^{24}+\frac{130839}{54476800}a^{23}+\frac{1637}{3891200}a^{22}+\frac{541}{136192}a^{21}+\frac{6493}{972800}a^{20}-\frac{276873}{27238400}a^{19}-\frac{13019}{1945600}a^{18}-\frac{5071}{340480}a^{17}-\frac{17411}{486400}a^{16}+\frac{5656697}{108953600}a^{15}+\frac{160251}{7782400}a^{14}+\frac{21627}{272384}a^{13}+\frac{282899}{1945600}a^{12}-\frac{165671}{851200}a^{11}-\frac{19557}{243200}a^{10}-\frac{34807}{85120}a^{9}-\frac{2283}{3800}a^{8}+\frac{2551}{6080}a^{7}+\frac{373}{1216}a^{6}+\frac{699}{608}a^{5}+\frac{1973}{760}a^{4}-\frac{15917}{13300}a^{3}-\frac{1847}{1900}a^{2}-\frac{2529}{1330}a-\frac{3113}{475}$ 431/27238400a^29 - 279/778240a^25 + 25121/13619200a^21 - 43167/6809600a^17 + 646943/27238400a^13 - 35531/851200a^9 + 253/1520a^5 - 6281/6650a, 543/13619200a^31 - 123/389120a^27 + 11163/6809600a^23 - 23721/3404800a^19 + 297279/13619200a^15 - 312479/3404800a^11 + 5777/12160a^7 - 18209/26600a^3, 723/10895360a^30 - 3351/27238400a^29 - 743/1556480a^26 + 623/778240a^25 + 10989/5447680a^22 - 53201/13619200a^21 - 22707/2723840a^18 + 98687/6809600a^17 + 440067/10895360a^14 - 1407303/27238400a^13 - 2833/21280a^10 + 379807/1702400a^9 + 2929/6080a^6 - 2339/3040a^5 - 2311/2660a^2 + 3818/3325a - 1, 1/77824a^28 + 73/389120a^24 - 33/38912a^20 - 453/97280a^16 + 7253/389120a^12 + 237/6080a^8 - 59/304a^4 - 12/95, 3443/27238400a^31 + 559/2723840a^30 - 4943/13619200a^29 - 863/778240a^27 - 491/389120a^26 + 9/4096a^25 + 60463/13619200a^23 + 8397/1361920a^22 - 68113/6809600a^21 - 131821/6809600a^19 - 15691/680960a^18 + 133871/3404800a^17 + 1978579/27238400a^15 + 59251/544768a^14 - 2183999/13619200a^13 - 2180579/6809600a^11 - 127317/340480a^10 + 142729/212800a^9 + 181/160a^7 + 8589/6080a^6 - 7101/3040a^5 - 68367/26600a^3 - 5647/2660a^2 + 13598/3325a, 321/3404800a^31 - 1/28672a^30 - 39/194560a^28 - 61/97280a^27 + 13/20480a^26 + 221/194560a^24 + 3441/1702400a^23 - 45/14336a^22 - 77/19456a^20 - 7347/851200a^19 + 87/7168a^18 + 1071/48640a^16 + 138433/3404800a^15 - 5861/143360a^14 - 15959/194560a^12 - 111043/851200a^11 + 6633/35840a^10 + 659/2432a^8 + 1211/3040a^7 - 33/40a^6 - 1649/1520a^4 - 8237/26600a^3 + 51/28a^2 + 42/19, 599/108953600a^31 + 159/3404800a^29 + 3/194560a^28 + 113/3112960a^27 - 3/5120a^25 - 169/194560a^24 - 47231/54476800a^23 + 2797/851200a^21 + 117/19456a^20 + 99857/27238400a^19 - 2207/212800a^17 - 179/9728a^16 - 505593/108953600a^15 + 144167/3404800a^13 + 14931/194560a^12 + 94887/6809600a^11 - 429899/1702400a^9 - 9529/24320a^8 - 19/128a^7 + 2987/3040a^5 + 377/380a^4 + 6711/26600a^3 - 5701/3325a - 174/95, 6367/108953600a^31 + 43/851200a^30 + 67/425600a^28 - 1511/3112960a^27 - 3/6080a^26 - 21/12160a^24 + 103297/54476800a^23 + 1511/851200a^22 + 39/5600a^20 - 224159/27238400a^19 - 3457/425600a^18 - 383/13300a^16 + 3490511/108953600a^15 + 1061/44800a^14 + 50571/425600a^12 - 211571/1702400a^11 - 98251/851200a^10 - 112047/212800a^8 + 1219/3040a^7 + 1947/6080a^6 + 1381/760a^4 - 10851/13300a^3 - 10147/13300a^2 - 16059/3325, 319/54476800a^30 - 103/851200a^28 - 967/1556480a^26 - 7/24320a^24 + 88289/27238400a^22 + 521/212800a^20 - 143743/13619200a^18 - 59/13300a^16 + 157173/2867200a^14 + 24561/851200a^12 - 173897/851200a^10 - 60577/425600a^8 + 5237/6080a^6 + 577/760a^4 - 31649/13300a^2 - 13372/3325, 6367/108953600a^31 + 543/2867200a^30 - 3351/13619200a^28 - 1511/3112960a^27 - 1989/1556480a^26 + 623/389120a^24 + 103297/54476800a^23 + 161347/27238400a^22 - 53201/6809600a^20 - 224159/27238400a^19 - 310909/13619200a^18 + 98687/3404800a^16 + 3490511/108953600a^15 + 5014941/54476800a^14 - 1407303/13619200a^12 - 211571/1702400a^11 - 606447/1702400a^10 + 379807/851200a^8 + 1219/3040a^7 + 7607/6080a^6 - 2339/1520a^4 - 10851/13300a^3 - 26827/13300a^2 + 10961/3325, 6911/108953600a^31 - 887/6809600a^29 - 519/3112960a^27 + 219/194560a^25 + 21601/54476800a^23 - 17207/3404800a^21 - 83327/27238400a^19 + 37549/1702400a^17 + 1414383/108953600a^15 - 458551/6809600a^13 - 52743/1702400a^11 + 508981/1702400a^9 + 621/12160a^7 - 1931/1520a^5 + 8539/26600a^3 + 7089/3325a, 13753/108953600a^31 + 2329/10895360a^30 - 349/1089536a^29 - 2437/6809600a^28 - 2961/3112960a^27 - 2117/1556480a^26 + 1773/778240a^25 + 557/194560a^24 + 192823/54476800a^23 + 5939/1089536a^22 - 24079/2723840a^21 - 37187/3404800a^20 - 421961/27238400a^19 - 57457/2723840a^18 + 46353/1361920a^17 + 86509/1702400a^16 + 5897929/108953600a^15 + 937449/10895360a^14 - 785377/5447680a^13 - 1367701/6809600a^12 - 726513/3404800a^11 - 257937/680960a^10 + 49047/85120a^9 + 339419/425600a^8 + 9639/12160a^7 + 865/608a^6 - 6817/3040a^5 - 4411/1520a^4 - 12519/13300a^3 - 2677/1330a^2 + 5083/1330a + 13434/3325, 503/5447680a^31 + 1901/7782400a^30 - 543/2723840a^29 - 3603/6809600a^28 - 239/778240a^27 - 3299/1556480a^26 + 79/77824a^25 + 899/194560a^24 + 607/544768a^23 + 37331/3891200a^22 - 1233/272384a^21 - 70673/3404800a^20 - 733/272384a^19 - 79117/1945600a^18 + 2535/136192a^17 + 149031/1702400a^16 + 1515/1089536a^15 + 1307453/7782400a^14 - 166127/2723840a^13 - 137521/358400a^12 + 2913/272384a^11 - 9661/15200a^10 + 65943/340480a^9 + 1298187/851200a^8 - 651/12160a^7 + 2739/1216a^6 - 89/160a^5 - 8383/1520a^4 + 569/665a^3 - 2019/475a^2 - 278/665a + 36111/3325, 647/3891200a^31 - 1893/10895360a^30 - 2131/5447680a^29 + 319/680960a^28 - 1109/778240a^27 + 2209/1556480a^26 + 571/155648a^25 - 21/5120a^24 + 11827/1945600a^23 - 39699/5447680a^22 - 39629/2723840a^21 + 6431/340480a^20 - 25609/972800a^19 + 73117/2723840a^18 + 84051/1361920a^17 - 2353/34048a^16 + 367591/3891200a^15 - 1190389/10895360a^14 - 1345411/5447680a^13 + 205439/680960a^12 - 400791/972800a^11 + 58001/136192a^10 + 87133/85120a^9 - 42025/34048a^8 + 19533/12160a^7 - 2669/1520a^6 - 65/19a^5 + 6381/1520a^4 - 1546/475a^3 + 10841/2660a^2 + 4717/665a - 1249/133, 4329/108953600a^31 - 53/7782400a^30 + 183/1361920a^29 + 483/1945600a^28 - 1633/3112960a^27 + 123/1556480a^26 - 203/194560a^25 - 717/389120a^24 + 130839/54476800a^23 + 1637/3891200a^22 + 541/136192a^21 + 6493/972800a^20 - 276873/27238400a^19 - 13019/1945600a^18 - 5071/340480a^17 - 17411/486400a^16 + 5656697/108953600a^15 + 160251/7782400a^14 + 21627/272384a^13 + 282899/1945600a^12 - 165671/851200a^11 - 19557/243200a^10 - 34807/85120a^9 - 2283/3800a^8 + 2551/6080a^7 + 373/1216a^6 + 699/608a^5 + 1973/760a^4 - 15917/13300a^3 - 1847/1900a^2 - 2529/1330a - 3113/475 (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:	$ 70769362065553.12 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 70769362065553.12 \cdot 558}{8\cdot\sqrt{11242761639679742219392447654740052674625150753177600000000}}\cr\approx \mathstrut & 0.274682432590373 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^32 - 11*x^28 + 62*x^24 - 260*x^20 + 1041*x^16 - 4160*x^12 + 15872*x^8 - 45056*x^4 + 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 - 11*x^28 + 62*x^24 - 260*x^20 + 1041*x^16 - 4160*x^12 + 15872*x^8 - 45056*x^4 + 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 - 11*x^28 + 62*x^24 - 260*x^20 + 1041*x^16 - 4160*x^12 + 15872*x^8 - 45056*x^4 + 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 - 11*x^28 + 62*x^24 - 260*x^20 + 1041*x^16 - 4160*x^12 + 15872*x^8 - 45056*x^4 + 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

$D_4^2:C_2^3$ (as 32T12882):

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 512

The 80 conjugacy class representatives for $D_4^2:C_2^3$

Character table for $D_4^2:C_2^3$

Intermediate fields

$\Q(\sqrt{-2}) $, $\Q(\sqrt{29}) $, $\Q(\sqrt{-29}) $, $\Q(\sqrt{2}) $, $\Q(\sqrt{-58}) $, $\Q(\sqrt{58}) $, $\Q(\sqrt{-1}) $, 4.4.269120.1, 4.0.67280.1, 4.4.4205.1, 4.0.269120.1, $\Q(i, \sqrt{58})$, $\Q(\sqrt{-2}, \sqrt{29})$, $\Q(\sqrt{2}, \sqrt{-29})$, $\Q(\sqrt{-2}, \sqrt{-29})$, $\Q(\sqrt{2}, \sqrt{29})$, $\Q(\zeta_{8})$, $\Q(i, \sqrt{29})$, 8.0.4968649025.1, 8.8.20351586406400.1, 8.0.20351586406400.1, 8.8.1271974150400.1, 8.0.46352367616.1, 8.0.1158809190400.13, 8.0.72425574400.8, 8.8.72425574400.1, 8.0.1158809190400.20, 8.0.1158809190400.9, 8.0.4526598400.4, 16.0.1342838739755503452160000.1, 16.0.414187069257165265960960000.2, 16.0.106031889729834308086005760000.1, 16.0.414187069257165265960960000.1, 16.16.106031889729834308086005760000.1, 16.0.1617918239285801820160000.1, 16.0.106031889729834308086005760000.2

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]

Sibling fields

Degree 32 siblings:	data not computed
Minimal sibling:	not computed

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.8.0.1}{8} }^{4}$	R	${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$	${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{8}$	${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{8}$	${\href{/padicField/17.4.0.1}{4} }^{8}$	${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$	${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$	R	${\href{/padicField/31.4.0.1}{4} }^{8}$	${\href{/padicField/37.8.0.1}{8} }^{4}$	${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$	${\href{/padicField/43.4.0.1}{4} }^{8}$	${\href{/padicField/47.8.0.1}{8} }^{4}$	${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{8}$	${\href{/padicField/59.4.0.1}{4} }^{8}$

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		Deg $16$	$4$	$4$	$32$
$2$ 2		Deg $16$	$4$	$4$	$32$
$5$ 5	5.2.0.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.2.0.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.2.0.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.2.0.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.2.0.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.2.0.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.2.0.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.2.0.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$29$ 29	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$281$ 281	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{281}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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L-functions

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Varieties

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