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Properties

Label 16.0.29233126977...0625.8
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 149^{12}$
Root discriminant $142.60$
Ramified primes $5, 149$
Class number $200$ (GRH)
Class group $[2, 10, 10]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9114300768451, 1006440846256, -118330534131, 9061447011, -11591574241, -563384901, 155539363, -37933807, 18367581, 998855, -3437, -21267, -8761, 345, 15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 15*x^14 + 345*x^13 - 8761*x^12 - 21267*x^11 - 3437*x^10 + 998855*x^9 + 18367581*x^8 - 37933807*x^7 + 155539363*x^6 - 563384901*x^5 - 11591574241*x^4 + 9061447011*x^3 - 118330534131*x^2 + 1006440846256*x + 9114300768451)
 
gp: K = bnfinit(x^16 - 2*x^15 + 15*x^14 + 345*x^13 - 8761*x^12 - 21267*x^11 - 3437*x^10 + 998855*x^9 + 18367581*x^8 - 37933807*x^7 + 155539363*x^6 - 563384901*x^5 - 11591574241*x^4 + 9061447011*x^3 - 118330534131*x^2 + 1006440846256*x + 9114300768451, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 15 x^{14} + 345 x^{13} - 8761 x^{12} - 21267 x^{11} - 3437 x^{10} + 998855 x^{9} + 18367581 x^{8} - 37933807 x^{7} + 155539363 x^{6} - 563384901 x^{5} - 11591574241 x^{4} + 9061447011 x^{3} - 118330534131 x^{2} + 1006440846256 x + 9114300768451 \)

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:  \(29233126977165462000179004150390625=5^{12}\cdot 149^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $142.60$
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:  $5, 149$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5001018884794} a^{14} + \frac{108872261785}{5001018884794} a^{13} + \frac{2312616314035}{5001018884794} a^{12} + \frac{356961241559}{5001018884794} a^{11} + \frac{413963410225}{5001018884794} a^{10} - \frac{1269400317921}{5001018884794} a^{9} - \frac{1505243534343}{5001018884794} a^{8} + \frac{2031733093623}{5001018884794} a^{7} + \frac{2302034351149}{5001018884794} a^{6} + \frac{825061574773}{5001018884794} a^{5} - \frac{1294141830653}{5001018884794} a^{4} + \frac{1246387242681}{5001018884794} a^{3} - \frac{455614759709}{5001018884794} a^{2} + \frac{1854506275015}{5001018884794} a + \frac{188275348111}{5001018884794}$, $\frac{1}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{15} + \frac{198190773700796140764034916920175893513334304757834447871800572975}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{14} - \frac{36373701360729868069247636536766272213831302178107574614006416457255601384298527}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{13} + \frac{27944401237937505738947155396249969157992243701026433043647290881089554778955113}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{12} - \frac{9934574947612648880802031034943799380895386847590088378045834630026876162808391}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{11} - \frac{19164661974876694014545318684784750443542534837192597464526593203853560561278833}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{10} - \frac{20449323768492464457730437785815895934747853654410288102329225508859175127065967}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{9} - \frac{15323094202772919529716562389306793248064301048899413944222972766575527747839425}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{8} + \frac{13458956790348885960279138053794652521846024586111634319468769409896810456619553}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{7} + \frac{9692352839770037208538182216749668340213697552377505622156910821344306369513323}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{6} - \frac{1551178316969794541360509513079945299705881936831945830471531243463127728988405}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{5} - \frac{424961905857367935518155016503249475722203032251532403504100092152474452818255}{3000648033337628074569016557563413369964650749503104240182356159708959299248954} a^{4} - \frac{8332525115724227669129552167561120876453942990044253403921882636597128426825645}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{3} + \frac{3588382524651609664809969309323118853017727432261884021775648431298858264890141}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a^{2} + \frac{33847677864028466213324692712235037637427635132324317887831378861141212289240037}{93020089033466470311639513284465814468904173234596231445653040950977738276717574} a - \frac{4731138439401901537327474220158735553497604653675624394244797750835886644662328}{46510044516733235155819756642232907234452086617298115722826520475488869138358787}$

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

Class group and class number

$C_{2}\times C_{10}\times C_{10}$, which has order $200$ (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:  \( 119396488.81 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{745}) \), \(\Q(\sqrt{149}) \), \(\Q(\sqrt{5}) \), 4.4.413493625.2, \(\Q(\sqrt{5}, \sqrt{149})\), 4.4.413493625.1, 4.0.111005.1 x2, 4.0.3725.1 x2, 8.8.170976977915640625.1, 8.0.308052750625.1, 8.0.170976977915640625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$149$149.4.3.1$x^{4} - 149$$4$$1$$3$$C_4$$[\ ]_{4}$
149.4.3.1$x^{4} - 149$$4$$1$$3$$C_4$$[\ ]_{4}$
149.4.3.1$x^{4} - 149$$4$$1$$3$$C_4$$[\ ]_{4}$
149.4.3.1$x^{4} - 149$$4$$1$$3$$C_4$$[\ ]_{4}$