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Properties

Label 16.0.57606497003...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $54.33$
Ramified primes $2, 3, 5, 11$
Class number $1024$ (GRH)
Class group $[4, 4, 4, 16]$ (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![1060195, -570320, 760490, -413680, 296066, -123900, 70602, -28016, 11499, -4324, 2154, -672, 112, -28, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 + 112*x^12 - 672*x^11 + 2154*x^10 - 4324*x^9 + 11499*x^8 - 28016*x^7 + 70602*x^6 - 123900*x^5 + 296066*x^4 - 413680*x^3 + 760490*x^2 - 570320*x + 1060195)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 28*x^13 + 112*x^12 - 672*x^11 + 2154*x^10 - 4324*x^9 + 11499*x^8 - 28016*x^7 + 70602*x^6 - 123900*x^5 + 296066*x^4 - 413680*x^3 + 760490*x^2 - 570320*x + 1060195, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 28 x^{13} + 112 x^{12} - 672 x^{11} + 2154 x^{10} - 4324 x^{9} + 11499 x^{8} - 28016 x^{7} + 70602 x^{6} - 123900 x^{5} + 296066 x^{4} - 413680 x^{3} + 760490 x^{2} - 570320 x + 1060195 \)

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:  \(5760649700315136000000000000=2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.33$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1320=2^{3}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(67,·)$, $\chi_{1320}(769,·)$, $\chi_{1320}(529,·)$, $\chi_{1320}(131,·)$, $\chi_{1320}(659,·)$, $\chi_{1320}(857,·)$, $\chi_{1320}(353,·)$, $\chi_{1320}(1123,·)$, $\chi_{1320}(593,·)$, $\chi_{1320}(617,·)$, $\chi_{1320}(43,·)$, $\chi_{1320}(419,·)$, $\chi_{1320}(241,·)$, $\chi_{1320}(307,·)$, $\chi_{1320}(1211,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{40} a^{12} + \frac{1}{10} a^{11} + \frac{3}{40} a^{10} - \frac{1}{10} a^{8} + \frac{7}{40} a^{6} + \frac{1}{5} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{40} a^{13} - \frac{3}{40} a^{11} - \frac{1}{20} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{3}{40} a^{7} - \frac{1}{4} a^{6} + \frac{1}{10} a^{5} - \frac{1}{10} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{846118375333160} a^{14} - \frac{7}{846118375333160} a^{13} - \frac{14275073587}{1572710734820} a^{12} + \frac{46079937538927}{846118375333160} a^{11} + \frac{249408017891}{846118375333160} a^{10} - \frac{293639931603}{423059187666580} a^{9} - \frac{78834864998093}{846118375333160} a^{8} - \frac{20375923233069}{846118375333160} a^{7} + \frac{80727512890443}{846118375333160} a^{6} + \frac{40646921045147}{423059187666580} a^{5} - \frac{24037306376473}{169223675066632} a^{4} - \frac{17466710993385}{169223675066632} a^{3} - \frac{6363952881195}{84611837533316} a^{2} + \frac{34057394561573}{169223675066632} a + \frac{13658367148125}{169223675066632}$, $\frac{1}{208775478521580564200} a^{15} + \frac{24673}{41755095704316112840} a^{14} + \frac{777959700204847257}{104387739260790282100} a^{13} - \frac{622201692257940201}{208775478521580564200} a^{12} - \frac{5596618136287869151}{208775478521580564200} a^{11} - \frac{935065271964055957}{20877547852158056420} a^{10} + \frac{1652829558912164489}{208775478521580564200} a^{9} - \frac{25271937474139177977}{208775478521580564200} a^{8} - \frac{38859979357677419107}{208775478521580564200} a^{7} - \frac{8974893909938762571}{104387739260790282100} a^{6} - \frac{47105260347808659919}{208775478521580564200} a^{5} - \frac{24568847164644172837}{208775478521580564200} a^{4} - \frac{87761030026981779}{5219386963039514105} a^{3} - \frac{8442829635966609607}{41755095704316112840} a^{2} - \frac{105549967172218673}{41755095704316112840} a + \frac{255550818225308819}{5219386963039514105}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{16}$, which has order $1024$ (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:  \( 19771.344992359085 \) (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

\(\Q(\sqrt{-66}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-330}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{6}, \sqrt{-11})\), \(\Q(\sqrt{5}, \sqrt{-66})\), \(\Q(\sqrt{30}, \sqrt{-55})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-55})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{-11}, \sqrt{30})\), 4.4.8000.1, 4.0.136125.2, \(\Q(\zeta_{15})^+\), 4.0.968000.5, 8.0.3035957760000.13, 8.0.75898944000000.138, 8.0.75898944000000.148, 8.8.5184000000.2, 8.0.75898944000000.116, 8.0.937024000000.3, 8.0.18530015625.1

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$