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Properties

Label 16.0.37447080302...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 97^{4}\cdot 101^{8}$
Root discriminant $70.52$
Ramified primes $5, 97, 101$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group 16T1275

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, -1375, 5200, -7575, 17200, -7300, 10445, 365, -2715, -4305, 2603, -275, -135, 93, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 4*x^14 + 93*x^13 - 135*x^12 - 275*x^11 + 2603*x^10 - 4305*x^9 - 2715*x^8 + 365*x^7 + 10445*x^6 - 7300*x^5 + 17200*x^4 - 7575*x^3 + 5200*x^2 - 1375*x + 125)
 
gp: K = bnfinit(x^16 - x^15 - 4*x^14 + 93*x^13 - 135*x^12 - 275*x^11 + 2603*x^10 - 4305*x^9 - 2715*x^8 + 365*x^7 + 10445*x^6 - 7300*x^5 + 17200*x^4 - 7575*x^3 + 5200*x^2 - 1375*x + 125, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 4 x^{14} + 93 x^{13} - 135 x^{12} - 275 x^{11} + 2603 x^{10} - 4305 x^{9} - 2715 x^{8} + 365 x^{7} + 10445 x^{6} - 7300 x^{5} + 17200 x^{4} - 7575 x^{3} + 5200 x^{2} - 1375 x + 125 \)

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:  \(374470803028447596341437890625=5^{8}\cdot 97^{4}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.52$
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:  $5, 97, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4}$, $\frac{1}{15} a^{12} + \frac{1}{15} a^{11} + \frac{1}{15} a^{10} + \frac{7}{15} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{15} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{75} a^{13} - \frac{1}{75} a^{12} - \frac{4}{75} a^{11} - \frac{2}{75} a^{10} + \frac{2}{15} a^{8} + \frac{28}{75} a^{7} - \frac{7}{15} a^{5} - \frac{7}{15} a^{4} + \frac{7}{15} a^{3} - \frac{1}{3}$, $\frac{1}{75} a^{14} - \frac{1}{75} a^{11} + \frac{1}{25} a^{10} - \frac{1}{15} a^{9} + \frac{23}{75} a^{8} + \frac{6}{25} a^{7} + \frac{2}{15} a^{6} + \frac{4}{15} a^{5} - \frac{1}{3} a^{4} + \frac{7}{15} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{95257932779188077264710475} a^{15} + \frac{325220923553350651259918}{95257932779188077264710475} a^{14} + \frac{74951174057772826407344}{95257932779188077264710475} a^{13} + \frac{77581513002185615050424}{3810317311167523090588419} a^{12} - \frac{4903172899042411937309291}{95257932779188077264710475} a^{11} - \frac{6005592066027956514842104}{95257932779188077264710475} a^{10} + \frac{805685153062016320846438}{95257932779188077264710475} a^{9} + \frac{11660012647348643784257684}{31752644259729359088236825} a^{8} - \frac{30805963828921998963734224}{95257932779188077264710475} a^{7} - \frac{432842974358474299610762}{3810317311167523090588419} a^{6} - \frac{6153667325848409342206208}{19051586555837615452942095} a^{5} - \frac{347741592688007686555453}{19051586555837615452942095} a^{4} + \frac{2532512300693618873822293}{6350528851945871817647365} a^{3} + \frac{1393040523770102865443902}{3810317311167523090588419} a^{2} + \frac{938435979543474076208578}{3810317311167523090588419} a + \frac{640367415564533441325689}{3810317311167523090588419}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 22577725.8153 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1275:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1275
Character table for t16n1275 is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.4947485.1, 8.0.122388039126125.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101Data not computed