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Properties

Label 16.8.34182052039...7929.3
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 41^{14}$
Root discriminant $607.24$
Ramified primes $37, 41$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![180850298131, 74815544047, -151944870564, -24006418810, 43477867525, -1928515073, -2811627137, 1085395296, -31765557, -92178988, -14781949, -2073739, -191213, -6068, -389, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 389*x^14 - 6068*x^13 - 191213*x^12 - 2073739*x^11 - 14781949*x^10 - 92178988*x^9 - 31765557*x^8 + 1085395296*x^7 - 2811627137*x^6 - 1928515073*x^5 + 43477867525*x^4 - 24006418810*x^3 - 151944870564*x^2 + 74815544047*x + 180850298131)
 
gp: K = bnfinit(x^16 - 389*x^14 - 6068*x^13 - 191213*x^12 - 2073739*x^11 - 14781949*x^10 - 92178988*x^9 - 31765557*x^8 + 1085395296*x^7 - 2811627137*x^6 - 1928515073*x^5 + 43477867525*x^4 - 24006418810*x^3 - 151944870564*x^2 + 74815544047*x + 180850298131, 1)
 

Normalized defining polynomial

\( x^{16} - 389 x^{14} - 6068 x^{13} - 191213 x^{12} - 2073739 x^{11} - 14781949 x^{10} - 92178988 x^{9} - 31765557 x^{8} + 1085395296 x^{7} - 2811627137 x^{6} - 1928515073 x^{5} + 43477867525 x^{4} - 24006418810 x^{3} - 151944870564 x^{2} + 74815544047 x + 180850298131 \)

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:  \(341820520390299824133982419608164514148827929=37^{14}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $607.24$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{127} a^{14} - \frac{42}{127} a^{13} + \frac{16}{127} a^{12} + \frac{46}{127} a^{11} - \frac{5}{127} a^{10} - \frac{30}{127} a^{9} + \frac{16}{127} a^{8} - \frac{9}{127} a^{7} + \frac{34}{127} a^{6} - \frac{36}{127} a^{5} - \frac{35}{127} a^{4} - \frac{32}{127} a^{3} - \frac{39}{127} a^{2} - \frac{56}{127} a + \frac{25}{127}$, $\frac{1}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{15} + \frac{678801597855405477520990073548138472499017224717420125105042472655831188524653659389618377}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{14} - \frac{71937186829713111039492795898838490064541960248403623334183362675713941923069750028715292009}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{13} - \frac{38478817513589416136508814207961001157149014869891218207447075597173871888653409411724511994}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{12} - \frac{17730658881007591496923587457959842378681024608298019139988360631595118731720110811208133171}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{11} - \frac{74514480283955916199149095547121082505260089889952879385957200479677449795508778411993350581}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{10} + \frac{44873070666244054295007274347052973598456869350997151676340455991163136915464568440978317378}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{9} - \frac{541437421252366623903858057445149852946953729173725837123280291909723114613417706182043498}{1765884475766741439688993536985602806908407088422950620204799330405950898948019117853921057} a^{8} + \frac{101758894171631611745497950505021148366814222349734092929708191201050223405491042916264309457}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{7} + \frac{57162824916676611631350308801531595228819764998705266692196869458288632631399640821570377730}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{6} - \frac{87320676666362717743075923832436100273237484801063058220940854565764030763448873261592158907}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{5} + \frac{89674104243994778484805225406618620045932773685351743329237041300173641855467329516619567673}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{4} + \frac{6220470941218798633942346170188484173505872105134391876210140570715526866826109239586237900}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{3} - \frac{44982039201135142166735483534573336326983969045839966424479167833921034016756786618821672016}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a^{2} - \frac{56674502655715789416391568635836230690695812920584771394906734743470409988468544927978197017}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239} a - \frac{64876289229875391044787248175777343460949253430467195823283291609465996999805981653610860545}{224267328422376162840502179197171556477367700229714728766009514961555764166398427967447974239}$

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

Class group and class number

$C_{4}$, 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:  \( 3876120808100000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{1517}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{37}, \sqrt{41})\), 8.8.12187467896636600569.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.2$x^{8} - 1476$$8$$1$$7$$C_8$$[\ ]_{8}$