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Properties

Label 16.0.10591011471...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 401^{4}$
Root discriminant $42.32$
Ramified primes $2, 5, 401$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T268)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1637641, -3224374, 1535495, 778892, -772272, -36422, 177806, -26482, -18615, 3994, 2262, -958, 12, 48, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 3*x^14 + 48*x^13 + 12*x^12 - 958*x^11 + 2262*x^10 + 3994*x^9 - 18615*x^8 - 26482*x^7 + 177806*x^6 - 36422*x^5 - 772272*x^4 + 778892*x^3 + 1535495*x^2 - 3224374*x + 1637641)
 
gp: K = bnfinit(x^16 - 6*x^15 + 3*x^14 + 48*x^13 + 12*x^12 - 958*x^11 + 2262*x^10 + 3994*x^9 - 18615*x^8 - 26482*x^7 + 177806*x^6 - 36422*x^5 - 772272*x^4 + 778892*x^3 + 1535495*x^2 - 3224374*x + 1637641, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 3 x^{14} + 48 x^{13} + 12 x^{12} - 958 x^{11} + 2262 x^{10} + 3994 x^{9} - 18615 x^{8} - 26482 x^{7} + 177806 x^{6} - 36422 x^{5} - 772272 x^{4} + 778892 x^{3} + 1535495 x^{2} - 3224374 x + 1637641 \)

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:  \(105910114717696000000000000=2^{24}\cdot 5^{12}\cdot 401^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.32$
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:  $2, 5, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{10} a^{8} - \frac{2}{5} a^{7} + \frac{3}{10} a^{6} - \frac{2}{5} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{10}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{10} - \frac{1}{10} a^{9} + \frac{2}{5} a^{8} - \frac{1}{2} a^{7} + \frac{1}{5} a^{6} + \frac{3}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{10} a + \frac{1}{5}$, $\frac{1}{30} a^{14} - \frac{1}{30} a^{13} - \frac{1}{30} a^{12} - \frac{13}{30} a^{10} - \frac{1}{2} a^{9} + \frac{1}{15} a^{8} + \frac{11}{30} a^{7} - \frac{2}{5} a^{6} - \frac{1}{6} a^{5} + \frac{13}{30} a^{4} - \frac{1}{15} a^{3} + \frac{3}{10} a^{2} - \frac{3}{10} a - \frac{13}{30}$, $\frac{1}{69960508986104899193761699117322538030} a^{15} + \frac{39061258833233277122641573862500007}{69960508986104899193761699117322538030} a^{14} + \frac{1188502512718240405272808500690328361}{69960508986104899193761699117322538030} a^{13} - \frac{132676398592554163760463561436080603}{4664033932406993279584113274488169202} a^{12} - \frac{5666039869867419247818567296294092189}{69960508986104899193761699117322538030} a^{11} - \frac{5767055786597777688305662299253070589}{23320169662034966397920566372440846010} a^{10} - \frac{742420244938321049530366615757357579}{34980254493052449596880849558661269015} a^{9} - \frac{11897598207821826448944506415628917848}{34980254493052449596880849558661269015} a^{8} + \frac{1637273695737607658172867161635223664}{11660084831017483198960283186220423005} a^{7} + \frac{2205746400997522125948584087402526239}{34980254493052449596880849558661269015} a^{6} + \frac{210766328148716050752494814806594461}{2256790612454996748185861261849114130} a^{5} - \frac{3926327668412455139640328740791795611}{13992101797220979838752339823464507606} a^{4} + \frac{4863241964346463331033396064140382013}{23320169662034966397920566372440846010} a^{3} - \frac{728924949301197183247968714367807615}{4664033932406993279584113274488169202} a^{2} + \frac{4725361238368692328231469558953213213}{13992101797220979838752339823464507606} a - \frac{8856373101525598124638060397332622967}{23320169662034966397920566372440846010}$

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:  \( \frac{115755767235538064283015813628}{2914292634595721869272752608403005} a^{15} - \frac{3427929588541936799482651683041}{17485755807574331215636515650418030} a^{14} - \frac{192237417696808528395475333267}{1748575580757433121563651565041803} a^{13} + \frac{16148281996553602826162684073199}{8742877903787165607818257825209015} a^{12} + \frac{7578769278721906823826575958731}{2914292634595721869272752608403005} a^{11} - \frac{626319429026673902107079896272121}{17485755807574331215636515650418030} a^{10} + \frac{143954398773297416480798491452846}{2914292634595721869272752608403005} a^{9} + \frac{3909367443487960616718142149427397}{17485755807574331215636515650418030} a^{8} - \frac{4384899630502428707076103135698002}{8742877903787165607818257825209015} a^{7} - \frac{9815084712274456929557188492540213}{5828585269191443738545505216806010} a^{6} + \frac{1493476333047334294114010377541372}{282028319477005342187685736297065} a^{5} + \frac{8734278989683726905676134582234367}{1748575580757433121563651565041803} a^{4} - \frac{227946296933265929459838395266659556}{8742877903787165607818257825209015} a^{3} - \frac{1485978703842960809515534956205523}{5828585269191443738545505216806010} a^{2} + \frac{187305223804878770702295501105042103}{2914292634595721869272752608403005} a - \frac{456004030750447101979705527624542588}{8742877903787165607818257825209015} \) (order $10$)
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:  \( 5693406.07584 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{5})\), 4.0.8000.2, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.64000000.2

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
401Data not computed