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Properties

Label 16.8.85171535827...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{6}\cdot 409^{2}\cdot 761^{6}$
Root discriminant $132.02$
Ramified primes $2, 5, 409, 761$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1127

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![315886199, -369718978, -960643601, -291554556, 162479515, 61595184, -13742081, -1516280, 755165, -205180, 13958, 1206, -1843, 400, -18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 18*x^14 + 400*x^13 - 1843*x^12 + 1206*x^11 + 13958*x^10 - 205180*x^9 + 755165*x^8 - 1516280*x^7 - 13742081*x^6 + 61595184*x^5 + 162479515*x^4 - 291554556*x^3 - 960643601*x^2 - 369718978*x + 315886199)
 
gp: K = bnfinit(x^16 - 4*x^15 - 18*x^14 + 400*x^13 - 1843*x^12 + 1206*x^11 + 13958*x^10 - 205180*x^9 + 755165*x^8 - 1516280*x^7 - 13742081*x^6 + 61595184*x^5 + 162479515*x^4 - 291554556*x^3 - 960643601*x^2 - 369718978*x + 315886199, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 18 x^{14} + 400 x^{13} - 1843 x^{12} + 1206 x^{11} + 13958 x^{10} - 205180 x^{9} + 755165 x^{8} - 1516280 x^{7} - 13742081 x^{6} + 61595184 x^{5} + 162479515 x^{4} - 291554556 x^{3} - 960643601 x^{2} - 369718978 x + 315886199 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8517153582795178291400802304000000=2^{24}\cdot 5^{6}\cdot 409^{2}\cdot 761^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.02$
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:  $2, 5, 409, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{23} a^{13} + \frac{7}{23} a^{12} + \frac{7}{23} a^{11} - \frac{8}{23} a^{9} + \frac{5}{23} a^{8} + \frac{8}{23} a^{7} - \frac{1}{23} a^{6} + \frac{3}{23} a^{5} + \frac{4}{23} a^{4} + \frac{6}{23} a^{3} - \frac{10}{23} a^{2} - \frac{6}{23} a + \frac{5}{23}$, $\frac{1}{23} a^{14} + \frac{4}{23} a^{12} - \frac{3}{23} a^{11} - \frac{8}{23} a^{10} - \frac{8}{23} a^{9} - \frac{4}{23} a^{8} - \frac{11}{23} a^{7} + \frac{10}{23} a^{6} + \frac{6}{23} a^{5} + \frac{1}{23} a^{4} - \frac{6}{23} a^{3} - \frac{5}{23} a^{2} + \frac{1}{23} a + \frac{11}{23}$, $\frac{1}{346054266863913496550075818739478738343702851800398877628814411453} a^{15} + \frac{130239116580110198655650858764890343819202736608366080432645363}{15045837689735369415220687771281684275813167469582559896904974411} a^{14} + \frac{6065325104733018990517395080652268534406936072962658584248964138}{346054266863913496550075818739478738343702851800398877628814411453} a^{13} - \frac{163822142750938767757923579433653071555777337706325764815223628770}{346054266863913496550075818739478738343702851800398877628814411453} a^{12} - \frac{4002599719993467491892907441156597229754213477952844940758167108}{49436323837701928078582259819925534049100407400056982518402058779} a^{11} - \frac{154486979266357095157228266451127117688820563146394543994601542}{346054266863913496550075818739478738343702851800398877628814411453} a^{10} + \frac{167874967681881628641278991841713806626919415237388756212291783210}{346054266863913496550075818739478738343702851800398877628814411453} a^{9} - \frac{157102157503935199544116332770404237322097145229968668587100685}{320124206164582327983418888750674133527939733395373614827765413} a^{8} + \frac{2704356425153979146977272737669401871449249066102232538900173595}{15045837689735369415220687771281684275813167469582559896904974411} a^{7} - \frac{9157825071986348729555650033377894787620641628017880019467471258}{49436323837701928078582259819925534049100407400056982518402058779} a^{6} + \frac{132538014162103891224118282964099140557920079587499516807453734}{1276953014257983382103600807156748111969383216975641614866473843} a^{5} - \frac{160694918290915006807753043662883369651635771067372013222313253626}{346054266863913496550075818739478738343702851800398877628814411453} a^{4} + \frac{148262872267762309629258682854251662204229044174866928318439891463}{346054266863913496550075818739478738343702851800398877628814411453} a^{3} + \frac{378811554131083679522324624615097907204787370529672732712720555}{1051836677397913363374090634466500724448944838299084734434086357} a^{2} - \frac{123028158444687521215191110405728977556892961051091531776389046534}{346054266863913496550075818739478738343702851800398877628814411453} a - \frac{128697444539702163315031054806285534476902222609751486189865022788}{346054266863913496550075818739478738343702851800398877628814411453}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 31602736714.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1127:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 58 conjugacy class representatives for t16n1127 are not computed
Character table for t16n1127 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.48704.1, 8.8.59301990400.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
409Data not computed
761Data not computed