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Properties

Label 32.0.368...256.1
Degree $32$
Signature $[0, 16]$
Discriminant $3.686\times 10^{50}$
Root discriminant \(38.04\)
Ramified primes $2,41,1277$
Class number $32$ (GRH)
Class group [4, 8] (GRH)
Galois group $C_2^6:S_4$ (as 32T96908)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 6*x^28 + 49*x^24 - 248*x^20 + 1072*x^16 - 3968*x^12 + 12544*x^8 - 24576*x^4 + 65536)
 
gp: K = bnfinit(y^32 - 6*y^28 + 49*y^24 - 248*y^20 + 1072*y^16 - 3968*y^12 + 12544*y^8 - 24576*y^4 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 6*x^28 + 49*x^24 - 248*x^20 + 1072*x^16 - 3968*x^12 + 12544*x^8 - 24576*x^4 + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 6*x^28 + 49*x^24 - 248*x^20 + 1072*x^16 - 3968*x^12 + 12544*x^8 - 24576*x^4 + 65536)
 

\( x^{32} - 6x^{28} + 49x^{24} - 248x^{20} + 1072x^{16} - 3968x^{12} + 12544x^{8} - 24576x^{4} + 65536 \) Copy content Toggle raw display

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:   \(368622913612148362854567389427658855145582683488256\) \(\medspace = 2^{64}\cdot 41^{4}\cdot 1277^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(38.04\)
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:  $2^{2}41^{1/2}1277^{1/2}\approx 915.266081530393$
Ramified primes:   \(2\), \(41\), \(1277\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}+\frac{1}{4}a^{4}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{7}-\frac{3}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{10}+\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{11}+\frac{1}{8}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{10}-\frac{1}{16}a^{8}+\frac{3}{16}a^{6}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{11}-\frac{1}{16}a^{9}+\frac{3}{16}a^{7}-\frac{1}{8}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{16}a^{14}-\frac{1}{8}a^{8}+\frac{3}{16}a^{6}-\frac{1}{8}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{16}-\frac{1}{16}a^{10}+\frac{3}{32}a^{8}-\frac{1}{16}a^{6}-\frac{1}{2}a^{5}+\frac{1}{8}a^{4}$, $\frac{1}{32}a^{17}-\frac{1}{16}a^{11}-\frac{1}{32}a^{9}+\frac{3}{16}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{64}a^{18}-\frac{1}{32}a^{14}+\frac{1}{64}a^{10}-\frac{3}{16}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{128}a^{19}-\frac{1}{64}a^{15}-\frac{7}{128}a^{11}-\frac{5}{32}a^{7}-\frac{1}{2}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{256}a^{20}-\frac{1}{128}a^{16}-\frac{7}{256}a^{12}-\frac{5}{64}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{5}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{512}a^{21}+\frac{3}{256}a^{17}+\frac{9}{512}a^{13}-\frac{1}{16}a^{11}-\frac{3}{128}a^{9}+\frac{3}{16}a^{7}-\frac{11}{32}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{512}a^{22}-\frac{1}{256}a^{18}-\frac{7}{512}a^{14}+\frac{3}{128}a^{10}-\frac{1}{32}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{1024}a^{23}-\frac{1}{512}a^{19}-\frac{7}{1024}a^{15}-\frac{13}{256}a^{11}-\frac{5}{64}a^{7}-\frac{1}{2}a^{5}+\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{2048}a^{24}-\frac{1}{1024}a^{22}+\frac{1}{1024}a^{20}-\frac{3}{512}a^{18}+\frac{17}{2048}a^{16}+\frac{23}{1024}a^{14}-\frac{1}{128}a^{12}+\frac{11}{256}a^{10}+\frac{1}{16}a^{8}-\frac{5}{64}a^{6}-\frac{1}{2}a^{5}-\frac{1}{32}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{4096}a^{25}-\frac{1}{2048}a^{23}+\frac{1}{2048}a^{21}-\frac{3}{1024}a^{19}+\frac{17}{4096}a^{17}+\frac{23}{2048}a^{15}+\frac{7}{256}a^{13}+\frac{27}{512}a^{11}-\frac{17}{128}a^{7}-\frac{5}{64}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{4096}a^{26}-\frac{1}{2048}a^{22}-\frac{1}{512}a^{20}-\frac{7}{4096}a^{18}-\frac{3}{256}a^{16}-\frac{13}{1024}a^{14}-\frac{9}{512}a^{12}+\frac{11}{256}a^{10}+\frac{3}{128}a^{8}-\frac{3}{32}a^{6}-\frac{5}{32}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{8192}a^{27}-\frac{1}{4096}a^{23}-\frac{1}{1024}a^{21}-\frac{7}{8192}a^{19}+\frac{5}{512}a^{17}+\frac{51}{2048}a^{15}-\frac{9}{1024}a^{13}+\frac{27}{512}a^{11}-\frac{1}{256}a^{9}+\frac{5}{64}a^{7}-\frac{5}{64}a^{5}-\frac{1}{2}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{491520}a^{28}-\frac{11}{245760}a^{24}+\frac{401}{491520}a^{20}-\frac{1}{256}a^{19}-\frac{1}{64}a^{17}-\frac{833}{61440}a^{16}-\frac{3}{128}a^{15}-\frac{1}{32}a^{13}-\frac{949}{30720}a^{12}+\frac{7}{256}a^{11}+\frac{3}{64}a^{9}-\frac{53}{3840}a^{8}+\frac{7}{64}a^{7}+\frac{5}{16}a^{5}+\frac{953}{1920}a^{4}-\frac{5}{16}a^{3}-\frac{1}{4}a+\frac{1}{120}$, $\frac{1}{983040}a^{29}-\frac{11}{491520}a^{25}+\frac{401}{983040}a^{21}-\frac{1}{128}a^{18}-\frac{833}{122880}a^{17}+\frac{1}{64}a^{14}-\frac{949}{61440}a^{13}+\frac{7}{128}a^{10}-\frac{53}{7680}a^{9}-\frac{1}{4}a^{7}+\frac{5}{32}a^{6}+\frac{1913}{3840}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{3}{8}a^{2}-\frac{119}{240}a-\frac{1}{2}$, $\frac{1}{3932160}a^{30}-\frac{1}{983040}a^{28}-\frac{11}{1966080}a^{26}+\frac{11}{491520}a^{24}+\frac{401}{3932160}a^{22}-\frac{401}{983040}a^{20}-\frac{1}{256}a^{19}-\frac{833}{491520}a^{18}+\frac{833}{122880}a^{16}-\frac{3}{128}a^{15}+\frac{6731}{245760}a^{14}+\frac{949}{61440}a^{12}-\frac{9}{256}a^{11}+\frac{1867}{30720}a^{10}-\frac{907}{7680}a^{8}-\frac{13}{64}a^{7}+\frac{473}{15360}a^{6}-\frac{1}{2}a^{5}+\frac{1447}{3840}a^{4}-\frac{1}{16}a^{3}-\frac{479}{960}a^{2}-\frac{1}{240}$, $\frac{1}{7864320}a^{31}-\frac{1}{1966080}a^{29}-\frac{11}{3932160}a^{27}+\frac{11}{983040}a^{25}+\frac{401}{7864320}a^{23}-\frac{401}{1966080}a^{21}-\frac{833}{983040}a^{19}+\frac{833}{245760}a^{17}+\frac{6731}{491520}a^{15}+\frac{949}{122880}a^{13}-\frac{1973}{61440}a^{11}-\frac{907}{15360}a^{9}-\frac{1}{8}a^{8}+\frac{6233}{30720}a^{7}-\frac{1}{4}a^{6}-\frac{473}{7680}a^{5}+\frac{3}{8}a^{4}-\frac{479}{1920}a^{3}+\frac{1}{4}a^{2}+\frac{239}{480}a-\frac{1}{2}$ Copy content Toggle raw display

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

$C_{4}\times C_{8}$, which has order $32$ (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:   \( -\frac{23}{1966080} a^{31} + \frac{13}{983040} a^{27} - \frac{583}{1966080} a^{23} + \frac{379}{245760} a^{19} - \frac{1093}{122880} a^{15} + \frac{199}{15360} a^{11} - \frac{319}{7680} a^{7} + \frac{37}{480} a^{3} \)  (order $8$) Copy content Toggle raw display
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:   $\frac{23}{1966080}a^{31}+\frac{7}{163840}a^{30}-\frac{13}{983040}a^{27}-\frac{37}{81920}a^{26}+\frac{583}{1966080}a^{23}+\frac{407}{163840}a^{22}-\frac{379}{245760}a^{19}-\frac{221}{20480}a^{18}+\frac{1093}{122880}a^{15}+\frac{477}{10240}a^{14}-\frac{199}{15360}a^{11}-\frac{151}{1280}a^{10}+\frac{319}{7680}a^{7}+\frac{211}{640}a^{6}-\frac{37}{480}a^{3}-\frac{23}{40}a^{2}-1$, $\frac{17}{1310720}a^{30}+\frac{17}{327680}a^{29}-\frac{3}{65536}a^{28}-\frac{187}{655360}a^{26}-\frac{27}{163840}a^{25}+\frac{1}{32768}a^{24}+\frac{1697}{1310720}a^{22}+\frac{417}{327680}a^{21}-\frac{51}{65536}a^{20}-\frac{1361}{163840}a^{18}-\frac{201}{40960}a^{17}-\frac{5}{8192}a^{16}+\frac{2747}{81920}a^{14}+\frac{707}{20480}a^{13}+\frac{47}{4096}a^{12}-\frac{1301}{10240}a^{10}-\frac{201}{2560}a^{9}-\frac{9}{512}a^{8}+\frac{1481}{5120}a^{6}+\frac{241}{1280}a^{5}+\frac{53}{256}a^{4}-\frac{263}{320}a^{2}-\frac{43}{80}a-\frac{11}{16}$, $\frac{7}{1572864}a^{31}-\frac{19}{1966080}a^{30}-\frac{29}{655360}a^{29}+\frac{7}{245760}a^{28}+\frac{19}{786432}a^{27}+\frac{209}{983040}a^{26}-\frac{1}{327680}a^{25}-\frac{17}{122880}a^{24}-\frac{649}{1572864}a^{23}-\frac{1859}{1966080}a^{22}+\frac{531}{655360}a^{21}+\frac{167}{245760}a^{20}+\frac{145}{196608}a^{19}+\frac{947}{245760}a^{18}-\frac{403}{81920}a^{17}-\frac{7}{3840}a^{16}-\frac{547}{98304}a^{15}-\frac{1769}{122880}a^{14}+\frac{1161}{40960}a^{13}+\frac{257}{15360}a^{12}+\frac{325}{12288}a^{11}+\frac{707}{15360}a^{10}-\frac{683}{5120}a^{9}-\frac{101}{1920}a^{8}-\frac{337}{6144}a^{7}-\frac{947}{7680}a^{6}+\frac{1043}{2560}a^{5}+\frac{101}{960}a^{4}+\frac{31}{384}a^{3}+\frac{101}{480}a^{2}-\frac{189}{160}a+\frac{7}{60}$, $\frac{139}{7864320}a^{31}-\frac{1}{24576}a^{30}-\frac{23}{655360}a^{29}+\frac{61}{491520}a^{28}-\frac{1049}{3932160}a^{27}+\frac{1}{6144}a^{26}+\frac{93}{327680}a^{25}-\frac{191}{245760}a^{24}+\frac{15419}{7864320}a^{23}-\frac{29}{24576}a^{22}-\frac{263}{655360}a^{21}+\frac{2381}{491520}a^{20}-\frac{11027}{983040}a^{19}+\frac{83}{12288}a^{18}+\frac{79}{81920}a^{17}-\frac{1493}{61440}a^{16}+\frac{23849}{491520}a^{15}-\frac{55}{3072}a^{14}+\frac{667}{40960}a^{13}+\frac{2531}{30720}a^{12}-\frac{9407}{61440}a^{11}+\frac{53}{768}a^{10}-\frac{481}{5120}a^{9}-\frac{983}{3840}a^{8}+\frac{11507}{30720}a^{7}-\frac{7}{48}a^{6}+\frac{681}{2560}a^{5}+\frac{1193}{1920}a^{4}-\frac{1181}{1920}a^{3}-\frac{1}{24}a^{2}-\frac{103}{160}a-\frac{119}{120}$, $\frac{83}{3932160}a^{31}-\frac{1}{983040}a^{29}-\frac{193}{1966080}a^{27}+\frac{11}{491520}a^{25}+\frac{3523}{3932160}a^{23}+\frac{559}{983040}a^{21}-\frac{2239}{491520}a^{19}-\frac{367}{122880}a^{17}+\frac{5113}{245760}a^{15}+\frac{1489}{61440}a^{13}-\frac{2059}{30720}a^{11}-\frac{877}{7680}a^{9}+\frac{3259}{15360}a^{7}+\frac{1747}{3840}a^{5}+\frac{83}{960}a^{3}-\frac{181}{240}a$, $\frac{161}{3932160}a^{31}+\frac{29}{393216}a^{30}-\frac{19}{491520}a^{29}-\frac{5}{32768}a^{28}-\frac{571}{1966080}a^{27}-\frac{79}{196608}a^{26}+\frac{89}{245760}a^{25}+\frac{15}{16384}a^{24}+\frac{6001}{3932160}a^{23}+\frac{1069}{393216}a^{22}-\frac{899}{491520}a^{21}-\frac{181}{32768}a^{20}-\frac{3733}{491520}a^{19}-\frac{673}{49152}a^{18}+\frac{557}{61440}a^{17}+\frac{107}{4096}a^{16}+\frac{5011}{245760}a^{15}+\frac{1231}{24576}a^{14}-\frac{719}{30720}a^{13}-\frac{203}{2048}a^{12}-\frac{1873}{30720}a^{11}-\frac{433}{3072}a^{10}+\frac{49}{480}a^{9}+\frac{79}{256}a^{8}+\frac{1513}{15360}a^{7}+\frac{541}{1536}a^{6}-\frac{437}{1920}a^{5}-\frac{93}{128}a^{4}+\frac{161}{960}a^{3}-\frac{31}{96}a^{2}+\frac{71}{120}a+\frac{15}{8}$, $\frac{91}{1966080}a^{31}-\frac{49}{786432}a^{30}+\frac{1}{16384}a^{29}-\frac{19}{983040}a^{28}-\frac{281}{983040}a^{27}+\frac{251}{393216}a^{26}-\frac{5}{8192}a^{25}+\frac{209}{491520}a^{24}+\frac{3851}{1966080}a^{23}-\frac{3137}{786432}a^{22}+\frac{73}{16384}a^{21}-\frac{1859}{983040}a^{20}-\frac{2423}{245760}a^{19}+\frac{2153}{98304}a^{18}-\frac{95}{4096}a^{17}+\frac{947}{122880}a^{16}+\frac{4541}{122880}a^{15}-\frac{4427}{49152}a^{14}+\frac{89}{1024}a^{13}-\frac{1769}{61440}a^{12}-\frac{1913}{15360}a^{11}+\frac{1805}{6144}a^{10}-\frac{9}{32}a^{9}+\frac{707}{7680}a^{8}+\frac{2663}{7680}a^{7}-\frac{2537}{3072}a^{6}+\frac{11}{16}a^{5}-\frac{947}{3840}a^{4}-\frac{119}{480}a^{3}+\frac{287}{192}a^{2}-a+\frac{341}{240}$, $\frac{7}{163840}a^{31}+\frac{49}{786432}a^{30}-\frac{13}{122880}a^{29}-\frac{19}{983040}a^{28}-\frac{17}{81920}a^{27}-\frac{251}{393216}a^{26}+\frac{19}{30720}a^{25}+\frac{209}{491520}a^{24}+\frac{247}{163840}a^{23}+\frac{3137}{786432}a^{22}-\frac{353}{122880}a^{21}-\frac{1859}{983040}a^{20}-\frac{39}{5120}a^{19}-\frac{2153}{98304}a^{18}+\frac{731}{61440}a^{17}+\frac{947}{122880}a^{16}+\frac{151}{5120}a^{15}+\frac{4427}{49152}a^{14}-\frac{323}{7680}a^{13}-\frac{1769}{61440}a^{12}-\frac{197}{2560}a^{11}-\frac{1805}{6144}a^{10}+\frac{89}{960}a^{9}+\frac{707}{7680}a^{8}+\frac{63}{320}a^{7}+\frac{2537}{3072}a^{6}-\frac{253}{960}a^{5}-\frac{947}{3840}a^{4}+\frac{1}{20}a^{3}-\frac{287}{192}a^{2}+\frac{1}{15}a+\frac{341}{240}$, $\frac{67}{3932160}a^{31}+\frac{111}{1310720}a^{30}+\frac{23}{983040}a^{29}+\frac{11}{983040}a^{28}+\frac{223}{1966080}a^{27}-\frac{261}{655360}a^{26}-\frac{133}{491520}a^{25}+\frac{119}{491520}a^{24}-\frac{1933}{3932160}a^{23}+\frac{3551}{1310720}a^{22}+\frac{2023}{983040}a^{21}-\frac{2309}{983040}a^{20}+\frac{2509}{491520}a^{19}-\frac{1663}{163840}a^{18}-\frac{1369}{122880}a^{17}+\frac{1457}{122880}a^{16}-\frac{5143}{245760}a^{15}+\frac{3221}{81920}a^{14}+\frac{2413}{61440}a^{13}-\frac{3719}{61440}a^{12}+\frac{2989}{30720}a^{11}-\frac{883}{10240}a^{10}-\frac{1459}{7680}a^{9}+\frac{2057}{7680}a^{8}-\frac{4189}{15360}a^{7}+\frac{1223}{5120}a^{6}+\frac{1459}{3840}a^{5}-\frac{2117}{3840}a^{4}+\frac{847}{960}a^{3}+\frac{111}{320}a^{2}-\frac{277}{240}a+\frac{611}{240}$, $\frac{323}{7864320}a^{31}+\frac{21}{1310720}a^{30}-\frac{3}{655360}a^{29}-\frac{3}{65536}a^{28}-\frac{1153}{3932160}a^{27}+\frac{89}{655360}a^{26}+\frac{33}{327680}a^{25}+\frac{1}{32768}a^{24}+\frac{12403}{7864320}a^{23}-\frac{539}{1310720}a^{22}-\frac{563}{655360}a^{21}-\frac{51}{65536}a^{20}-\frac{8299}{983040}a^{19}+\frac{507}{163840}a^{18}+\frac{419}{81920}a^{17}-\frac{5}{8192}a^{16}+\frac{16753}{491520}a^{15}-\frac{2009}{81920}a^{14}-\frac{633}{40960}a^{13}+\frac{47}{4096}a^{12}-\frac{5719}{61440}a^{11}+\frac{727}{10240}a^{10}+\frac{419}{5120}a^{9}-\frac{9}{512}a^{8}+\frac{9259}{30720}a^{7}-\frac{947}{5120}a^{6}-\frac{99}{2560}a^{5}+\frac{53}{256}a^{4}-\frac{1117}{1920}a^{3}+\frac{301}{320}a^{2}+\frac{77}{160}a-\frac{3}{16}$, $\frac{5}{196608}a^{31}+\frac{47}{1310720}a^{30}-\frac{13}{196608}a^{29}-\frac{5}{196608}a^{28}-\frac{7}{98304}a^{27}-\frac{37}{655360}a^{26}+\frac{47}{98304}a^{25}+\frac{7}{98304}a^{24}+\frac{85}{196608}a^{23}+\frac{287}{1310720}a^{22}-\frac{605}{196608}a^{21}-\frac{277}{196608}a^{20}-\frac{73}{24576}a^{19}+\frac{9}{163840}a^{18}+\frac{341}{24576}a^{17}+\frac{25}{24576}a^{16}+\frac{19}{12288}a^{15}-\frac{2043}{81920}a^{14}-\frac{791}{12288}a^{13}-\frac{223}{12288}a^{12}-\frac{13}{1536}a^{11}+\frac{789}{10240}a^{10}+\frac{221}{1536}a^{9}-\frac{59}{1536}a^{8}-\frac{35}{768}a^{7}-\frac{1529}{5120}a^{6}-\frac{317}{768}a^{5}+\frac{83}{768}a^{4}+\frac{5}{12}a^{3}+\frac{327}{320}a^{2}+\frac{11}{48}a-\frac{29}{48}$, $\frac{203}{3932160}a^{31}+\frac{47}{1310720}a^{30}+\frac{7}{491520}a^{29}+\frac{5}{196608}a^{28}-\frac{313}{1966080}a^{27}-\frac{37}{655360}a^{26}-\frac{77}{245760}a^{25}-\frac{7}{98304}a^{24}+\frac{4603}{3932160}a^{23}+\frac{287}{1310720}a^{22}+\frac{887}{491520}a^{21}+\frac{277}{196608}a^{20}-\frac{1579}{491520}a^{19}+\frac{9}{163840}a^{18}-\frac{551}{61440}a^{17}-\frac{25}{24576}a^{16}+\frac{1753}{245760}a^{15}-\frac{2043}{81920}a^{14}+\frac{917}{30720}a^{13}+\frac{223}{12288}a^{12}-\frac{439}{30720}a^{11}+\frac{789}{10240}a^{10}-\frac{251}{3840}a^{9}+\frac{59}{1536}a^{8}+\frac{499}{15360}a^{7}-\frac{1529}{5120}a^{6}+\frac{431}{1920}a^{5}-\frac{83}{768}a^{4}+\frac{203}{960}a^{3}+\frac{327}{320}a^{2}+\frac{37}{120}a+\frac{29}{48}$, $\frac{41}{2621440}a^{31}+\frac{17}{491520}a^{30}+\frac{11}{131072}a^{29}-\frac{1}{7680}a^{28}-\frac{131}{1310720}a^{27}-\frac{7}{245760}a^{26}+\frac{7}{65536}a^{25}+\frac{13}{30720}a^{24}+\frac{2361}{2621440}a^{23}-\frac{143}{491520}a^{22}+\frac{59}{131072}a^{21}-\frac{67}{15360}a^{20}-\frac{793}{327680}a^{19}+\frac{11}{15360}a^{18}+\frac{21}{16384}a^{17}+\frac{541}{30720}a^{16}+\frac{3011}{163840}a^{15}-\frac{653}{30720}a^{14}-\frac{119}{8192}a^{13}-\frac{551}{7680}a^{12}-\frac{1213}{20480}a^{11}+\frac{299}{3840}a^{10}+\frac{65}{1024}a^{9}+\frac{481}{1920}a^{8}+\frac{1313}{10240}a^{7}-\frac{329}{1920}a^{6}-\frac{93}{512}a^{5}-\frac{77}{120}a^{4}-\frac{359}{640}a^{3}+\frac{167}{120}a^{2}+\frac{27}{32}a+\frac{29}{30}$, $\frac{89}{1310720}a^{31}+\frac{189}{1310720}a^{30}+\frac{49}{327680}a^{29}+\frac{127}{983040}a^{28}-\frac{339}{655360}a^{27}-\frac{639}{655360}a^{26}-\frac{219}{163840}a^{25}-\frac{677}{491520}a^{24}+\frac{3689}{1310720}a^{23}+\frac{7309}{1310720}a^{22}+\frac{2369}{327680}a^{21}+\frac{5807}{983040}a^{20}-\frac{2617}{163840}a^{19}-\frac{4757}{163840}a^{18}-\frac{1537}{40960}a^{17}-\frac{4331}{122880}a^{16}+\frac{4899}{81920}a^{15}+\frac{8959}{81920}a^{14}+\frac{3219}{20480}a^{13}+\frac{8477}{61440}a^{12}-\frac{1637}{10240}a^{11}-\frac{3217}{10240}a^{10}-\frac{1137}{2560}a^{9}-\frac{2951}{7680}a^{8}+\frac{2817}{5120}a^{7}+\frac{4677}{5120}a^{6}+\frac{1617}{1280}a^{5}+\frac{3911}{3840}a^{4}-\frac{231}{320}a^{3}-\frac{451}{320}a^{2}-\frac{211}{80}a-\frac{593}{240}$, $\frac{13}{983040}a^{31}-\frac{239}{3932160}a^{30}+\frac{7}{163840}a^{29}+\frac{49}{983040}a^{28}-\frac{143}{491520}a^{27}+\frac{229}{1966080}a^{26}-\frac{37}{81920}a^{25}-\frac{299}{491520}a^{24}+\frac{2333}{983040}a^{23}-\frac{5599}{3932160}a^{22}+\frac{407}{163840}a^{21}+\frac{3329}{983040}a^{20}-\frac{1469}{122880}a^{19}+\frac{2647}{491520}a^{18}-\frac{221}{20480}a^{17}-\frac{2837}{122880}a^{16}+\frac{3323}{61440}a^{15}-\frac{4069}{245760}a^{14}+\frac{477}{10240}a^{13}+\frac{4739}{61440}a^{12}-\frac{1379}{7680}a^{11}+\frac{427}{30720}a^{10}-\frac{151}{1280}a^{9}-\frac{1697}{7680}a^{8}+\frac{1649}{3840}a^{7}-\frac{967}{15360}a^{6}+\frac{211}{640}a^{5}+\frac{2777}{3840}a^{4}-\frac{167}{240}a^{3}-\frac{479}{960}a^{2}-\frac{43}{40}a-\frac{191}{240}$ Copy content Toggle raw display (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:  \( 747032345387.1615 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 747032345387.1615 \cdot 32}{8\cdot\sqrt{368622913612148362854567389427658855145582683488256}}\cr\approx \mathstrut & 0.918303263736995 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 6*x^28 + 49*x^24 - 248*x^20 + 1072*x^16 - 3968*x^12 + 12544*x^8 - 24576*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 - 6*x^28 + 49*x^24 - 248*x^20 + 1072*x^16 - 3968*x^12 + 12544*x^8 - 24576*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 - 6*x^28 + 49*x^24 - 248*x^20 + 1072*x^16 - 3968*x^12 + 12544*x^8 - 24576*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 - 6*x^28 + 49*x^24 - 248*x^20 + 1072*x^16 - 3968*x^12 + 12544*x^8 - 24576*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

$C_2^6:S_4$ (as 32T96908):

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 1536
The 80 conjugacy class representatives for $C_2^6:S_4$
Character table for $C_2^6:S_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), 4.4.40864.1, \(\Q(\zeta_{8})\), 8.8.106871455744.1, 8.8.1095432421376.1, 8.0.26717863936.1, 8.0.4381729685504.1, 8.0.68464526336.1, 8.0.6679465984.1, 8.8.273858105344.1, 16.16.19199555036826982747734016.1, 16.0.11421508052841750593536.1, 16.0.19199555036826982747734016.1, 16.0.1199972189801686421733376.1, 16.0.19199555036826982747734016.2, 16.0.19199555036826982747734016.3, 16.0.74998261862605401358336.1

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 ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ ${\href{/padicField/5.6.0.1}{6} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{4}$ ${\href{/padicField/17.3.0.1}{3} }^{8}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.8.0.1}{8} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{8}$ ${\href{/padicField/37.6.0.1}{6} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ R ${\href{/padicField/43.2.0.1}{2} }^{16}$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.6.0.1}{6} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
\(41\) Copy content Toggle raw display $\Q_{41}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 35$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} + 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} + 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} + 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} + 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
\(1277\) Copy content Toggle raw display Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$