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Properties

Label 16.0.67467650588...3056.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 7^{8}\cdot 17^{8}$
Root discriminant $30.85$
Ramified primes $2, 7, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times S_4$ (as 16T61)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![784, -3136, 6272, -8288, 6316, -1736, 224, -1212, 2325, -1558, 562, -114, 79, -66, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 66*x^13 + 79*x^12 - 114*x^11 + 562*x^10 - 1558*x^9 + 2325*x^8 - 1212*x^7 + 224*x^6 - 1736*x^5 + 6316*x^4 - 8288*x^3 + 6272*x^2 - 3136*x + 784)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 66*x^13 + 79*x^12 - 114*x^11 + 562*x^10 - 1558*x^9 + 2325*x^8 - 1212*x^7 + 224*x^6 - 1736*x^5 + 6316*x^4 - 8288*x^3 + 6272*x^2 - 3136*x + 784, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 66 x^{13} + 79 x^{12} - 114 x^{11} + 562 x^{10} - 1558 x^{9} + 2325 x^{8} - 1212 x^{7} + 224 x^{6} - 1736 x^{5} + 6316 x^{4} - 8288 x^{3} + 6272 x^{2} - 3136 x + 784 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(674676505885957534253056=2^{24}\cdot 7^{8}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.85$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{112} a^{12} - \frac{1}{8} a^{11} + \frac{1}{56} a^{10} + \frac{17}{56} a^{9} - \frac{45}{112} a^{8} - \frac{3}{14} a^{7} + \frac{3}{28} a^{6} - \frac{3}{8} a^{5} - \frac{51}{112} a^{4} - \frac{19}{56} a^{3} + \frac{25}{56} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{112} a^{13} - \frac{13}{56} a^{11} + \frac{3}{56} a^{10} - \frac{17}{112} a^{9} + \frac{9}{56} a^{8} - \frac{11}{28} a^{7} - \frac{3}{8} a^{6} + \frac{33}{112} a^{5} + \frac{2}{7} a^{4} + \frac{11}{56} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{162848} a^{14} - \frac{99}{40712} a^{13} - \frac{27}{40712} a^{12} - \frac{12167}{81424} a^{11} - \frac{1997}{162848} a^{10} + \frac{5147}{11632} a^{9} + \frac{163}{81424} a^{8} + \frac{9871}{81424} a^{7} - \frac{71567}{162848} a^{6} - \frac{1647}{20356} a^{5} - \frac{10181}{40712} a^{4} - \frac{6103}{40712} a^{3} - \frac{7649}{20356} a^{2} - \frac{246}{727} a - \frac{515}{2908}$, $\frac{1}{121255800062201952} a^{15} + \frac{62165611147}{60627900031100976} a^{14} + \frac{8377048881295}{4330564287935784} a^{13} - \frac{167963586450799}{60627900031100976} a^{12} - \frac{15582755655806569}{121255800062201952} a^{11} - \frac{1161486314349943}{7578487503887622} a^{10} - \frac{2295131124222533}{20209300010366992} a^{9} - \frac{332170062280481}{8661128575871568} a^{8} - \frac{31914346428336595}{121255800062201952} a^{7} + \frac{19628128264042933}{60627900031100976} a^{6} - \frac{1588207692723587}{30313950015550488} a^{5} + \frac{15655101414095}{41697317765544} a^{4} - \frac{60799520617768}{1263081250647937} a^{3} + \frac{3742471129016815}{7578487503887622} a^{2} - \frac{8831154534393}{721760714655964} a + \frac{59216945897849}{1082641071983946}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{27347834471}{10424329441386} a^{15} + \frac{788656953185}{41697317765544} a^{14} - \frac{206830209181}{2978379840396} a^{13} + \frac{634887911342}{5212164720693} a^{12} - \frac{2599257882149}{20848658882772} a^{11} + \frac{9454715606303}{41697317765544} a^{10} - \frac{4584876596879}{3474776480462} a^{9} + \frac{64227188993047}{20848658882772} a^{8} - \frac{81438646965545}{20848658882772} a^{7} + \frac{32225862299885}{41697317765544} a^{6} - \frac{16532329235665}{20848658882772} a^{5} + \frac{21009474254452}{5212164720693} a^{4} - \frac{45646191795649}{3474776480462} a^{3} + \frac{67388158870694}{5212164720693} a^{2} - \frac{2312874247605}{248198320033} a + \frac{2648043261850}{744594960099} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 265528.345569 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_4$ (as 16T61):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $C_2\times S_4$
Character table for $C_2\times S_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), 4.4.32368.1, \(\Q(i, \sqrt{7})\), 8.8.821386940416.1, 8.0.16762998784.2, 8.0.51336683776.7

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

Sibling fields

Degree 6 siblings: data not computed
Degree 8 siblings: data not computed
Degree 12 siblings: data not computed
Degree 24 siblings: data 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$