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Properties

Label 16.0.10922431682...5344.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{42}\cdot 3^{8}\cdot 17^{6}\cdot 97^{2}\cdot 1291^{2}$
Root discriminant $134.09$
Ramified primes $2, 3, 17, 97, 1291$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![272446237993, 17297644568, 44818888652, -13121006632, 5621801606, 738308640, 315562164, -12843008, 12590434, -1036800, 80044, -44160, 7974, -264, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 264*x^13 + 7974*x^12 - 44160*x^11 + 80044*x^10 - 1036800*x^9 + 12590434*x^8 - 12843008*x^7 + 315562164*x^6 + 738308640*x^5 + 5621801606*x^4 - 13121006632*x^3 + 44818888652*x^2 + 17297644568*x + 272446237993)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 264*x^13 + 7974*x^12 - 44160*x^11 + 80044*x^10 - 1036800*x^9 + 12590434*x^8 - 12843008*x^7 + 315562164*x^6 + 738308640*x^5 + 5621801606*x^4 - 13121006632*x^3 + 44818888652*x^2 + 17297644568*x + 272446237993, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 264 x^{13} + 7974 x^{12} - 44160 x^{11} + 80044 x^{10} - 1036800 x^{9} + 12590434 x^{8} - 12843008 x^{7} + 315562164 x^{6} + 738308640 x^{5} + 5621801606 x^{4} - 13121006632 x^{3} + 44818888652 x^{2} + 17297644568 x + 272446237993 \)

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:  \(10922431682739176997092820825145344=2^{42}\cdot 3^{8}\cdot 17^{6}\cdot 97^{2}\cdot 1291^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.09$
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:  $2, 3, 17, 97, 1291$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{8} a^{8} - \frac{3}{8} a^{4} - \frac{1}{4} a^{2} + \frac{3}{8}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{11} - \frac{1}{8} a^{9} - \frac{3}{8} a^{5} - \frac{1}{4} a^{3} + \frac{3}{8} a$, $\frac{1}{1267208} a^{14} - \frac{61}{158401} a^{13} + \frac{15289}{633604} a^{12} - \frac{22978}{158401} a^{11} - \frac{163541}{1267208} a^{10} - \frac{8145}{316802} a^{9} + \frac{116}{1633} a^{8} - \frac{37617}{158401} a^{7} - \frac{292443}{1267208} a^{6} - \frac{7873}{158401} a^{5} + \frac{54893}{633604} a^{4} - \frac{56859}{158401} a^{3} + \frac{90815}{1267208} a^{2} - \frac{86831}{316802} a + \frac{11415}{158401}$, $\frac{1}{12014446930315861447712351034315219981298012408683164766786016459303169782385776} a^{15} - \frac{4504977468261442114266744905193054419203601902063425642234118121639435911}{12014446930315861447712351034315219981298012408683164766786016459303169782385776} a^{14} - \frac{662908832525164328182399401004171548600883191116222535743611742148700466324637}{12014446930315861447712351034315219981298012408683164766786016459303169782385776} a^{13} + \frac{662445917393923494327144867015085115688199645608939627803665674505012132115159}{12014446930315861447712351034315219981298012408683164766786016459303169782385776} a^{12} - \frac{794603716749022886311274452214156373872904872635426437513398761254906664478899}{12014446930315861447712351034315219981298012408683164766786016459303169782385776} a^{11} - \frac{2826928141807863601047779909468477240388436273441133827591990350995584606518187}{12014446930315861447712351034315219981298012408683164766786016459303169782385776} a^{10} + \frac{66760933806361769449677829260546331492645968319660047863114958967711814087123}{293035290983313693846642708154029755641414936797150360165512596568369994692336} a^{9} + \frac{2806047443976531810982066674670560704146162212050818651875596682930709536391703}{12014446930315861447712351034315219981298012408683164766786016459303169782385776} a^{8} - \frac{2465241285844100687934570278032590908201417804922519242519233264458273462380163}{12014446930315861447712351034315219981298012408683164766786016459303169782385776} a^{7} + \frac{1881607038299130669207000159933890930577264861612850356696134817253756710647965}{12014446930315861447712351034315219981298012408683164766786016459303169782385776} a^{6} - \frac{2089063242131666152258597386628369977013032401385321910434203562669826827756257}{12014446930315861447712351034315219981298012408683164766786016459303169782385776} a^{5} + \frac{11719621388443808606388282233277667313360241727975195388914278892400874087779}{293035290983313693846642708154029755641414936797150360165512596568369994692336} a^{4} - \frac{5354636852267469413054584831657242130595203284137825443473302132994962750517439}{12014446930315861447712351034315219981298012408683164766786016459303169782385776} a^{3} - \frac{5039208396793249358802329608923206290851128855857004532575575754280696377092783}{12014446930315861447712351034315219981298012408683164766786016459303169782385776} a^{2} - \frac{72510137509150875112775494723253926850768178280637046852356096187347590193209}{293035290983313693846642708154029755641414936797150360165512596568369994692336} a - \frac{1995695832563368748602812800075849420550044860024918768494273717918912352132509}{12014446930315861447712351034315219981298012408683164766786016459303169782385776}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 4673299084.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}, \sqrt{3})\), 4.0.1088.2, 4.0.39168.3, 8.0.1534132224.4

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.22.99$x^{8} + 4 x^{7} + 4 x^{6} + 6 x^{4} + 12 x^{2} + 14$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
2.8.20.58$x^{8} + 8 x^{6} + 64 x + 16$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{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$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
1291Data not computed