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Properties

Label 16.0.12192752578...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 11^{8}\cdot 13^{12}$
Root discriminant $75.92$
Ramified primes $5, 11, 13$
Class number $11600$ (GRH)
Class group $[10, 1160]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2784016, -2485508, 2445759, -1011818, 668076, -247816, 144027, -27084, 7277, -1492, 2683, -892, 61, 76, -4, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 4*x^14 + 76*x^13 + 61*x^12 - 892*x^11 + 2683*x^10 - 1492*x^9 + 7277*x^8 - 27084*x^7 + 144027*x^6 - 247816*x^5 + 668076*x^4 - 1011818*x^3 + 2445759*x^2 - 2485508*x + 2784016)
 
gp: K = bnfinit(x^16 - 6*x^15 - 4*x^14 + 76*x^13 + 61*x^12 - 892*x^11 + 2683*x^10 - 1492*x^9 + 7277*x^8 - 27084*x^7 + 144027*x^6 - 247816*x^5 + 668076*x^4 - 1011818*x^3 + 2445759*x^2 - 2485508*x + 2784016, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 4 x^{14} + 76 x^{13} + 61 x^{12} - 892 x^{11} + 2683 x^{10} - 1492 x^{9} + 7277 x^{8} - 27084 x^{7} + 144027 x^{6} - 247816 x^{5} + 668076 x^{4} - 1011818 x^{3} + 2445759 x^{2} - 2485508 x + 2784016 \)

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:  \(1219275257885198999941650390625=5^{12}\cdot 11^{8}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.92$
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:  $5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(715=5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{715}(1,·)$, $\chi_{715}(131,·)$, $\chi_{715}(584,·)$, $\chi_{715}(714,·)$, $\chi_{715}(463,·)$, $\chi_{715}(144,·)$, $\chi_{715}(593,·)$, $\chi_{715}(274,·)$, $\chi_{715}(408,·)$, $\chi_{715}(538,·)$, $\chi_{715}(177,·)$, $\chi_{715}(307,·)$, $\chi_{715}(441,·)$, $\chi_{715}(122,·)$, $\chi_{715}(571,·)$, $\chi_{715}(252,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{56} a^{11} - \frac{3}{56} a^{10} - \frac{3}{56} a^{9} - \frac{1}{14} a^{8} + \frac{1}{14} a^{7} + \frac{3}{56} a^{5} + \frac{3}{56} a^{4} - \frac{25}{56} a^{3} + \frac{3}{14} a^{2} + \frac{2}{7}$, $\frac{1}{224} a^{12} + \frac{9}{224} a^{10} + \frac{1}{28} a^{9} + \frac{13}{224} a^{8} - \frac{1}{14} a^{7} - \frac{25}{224} a^{6} + \frac{3}{56} a^{5} + \frac{47}{224} a^{4} + \frac{3}{8} a^{3} - \frac{69}{224} a^{2} + \frac{11}{56} a + \frac{3}{14}$, $\frac{1}{224} a^{13} + \frac{1}{224} a^{11} + \frac{1}{56} a^{10} + \frac{9}{224} a^{9} - \frac{3}{56} a^{8} - \frac{1}{224} a^{7} + \frac{3}{56} a^{6} + \frac{23}{224} a^{5} - \frac{3}{28} a^{4} + \frac{47}{224} a^{3} + \frac{11}{28} a^{2} - \frac{1}{28} a + \frac{3}{7}$, $\frac{1}{896} a^{14} + \frac{1}{896} a^{13} - \frac{1}{128} a^{11} + \frac{3}{224} a^{10} - \frac{3}{896} a^{9} - \frac{31}{448} a^{8} - \frac{11}{128} a^{7} + \frac{43}{224} a^{6} - \frac{23}{128} a^{5} - \frac{9}{56} a^{4} - \frac{293}{896} a^{3} - \frac{359}{896} a^{2} - \frac{31}{224} a + \frac{19}{56}$, $\frac{1}{57420362145900500412944470107776} a^{15} + \frac{16433854530788333001022115269}{57420362145900500412944470107776} a^{14} - \frac{8881596059671291614404277359}{7177545268237562551618058763472} a^{13} + \frac{121042418716030503328789471209}{57420362145900500412944470107776} a^{12} - \frac{812119425396751426955609519}{14355090536475125103236117526944} a^{11} + \frac{505047336824637849308665979805}{57420362145900500412944470107776} a^{10} - \frac{54843603858840931197358741005}{4101454438992892886638890721984} a^{9} + \frac{7110645028779603360626574629307}{57420362145900500412944470107776} a^{8} - \frac{1370598082234164211891726196259}{14355090536475125103236117526944} a^{7} + \frac{8456909507671142858204030705935}{57420362145900500412944470107776} a^{6} + \frac{573598951771745917749099766647}{7177545268237562551618058763472} a^{5} + \frac{1035362747351543563614296203419}{57420362145900500412944470107776} a^{4} + \frac{4499574045200848426512432746257}{57420362145900500412944470107776} a^{3} + \frac{1953905634666657785319142533187}{7177545268237562551618058763472} a^{2} + \frac{843536667239426206221790346415}{1794386317059390637904514690868} a - \frac{331705939032906280081682932399}{897193158529695318952257345434}$

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

Class group and class number

$C_{10}\times C_{1160}$, which has order $11600$ (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:  \( -1 \) (order $2$)
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:  \( 136143.5905281255 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-715}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-143}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}, \sqrt{-143})\), \(\Q(\sqrt{-11}, \sqrt{65})\), \(\Q(\sqrt{13}, \sqrt{-55})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{-55}, \sqrt{65})\), \(\Q(\sqrt{-11}, \sqrt{13})\), 4.0.33229625.1, 4.4.274625.2, 4.0.33229625.2, 4.4.274625.1, 8.0.261351000625.1, 8.0.1104207977640625.4, 8.0.1104207977640625.5, 8.0.1104207977640625.11, 8.8.75418890625.1, 8.0.1104207977640625.8, 8.0.1104207977640625.10

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$