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Properties

Label 16.0.37319027463...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 13^{12}$
Root discriminant $39.65$
Ramified primes $3, 5, 13$
Class number $52$ (GRH)
Class group $[2, 26]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15856, -27624, 26944, 10752, -7933, 1980, 1788, -3846, 1304, -294, 219, -96, 83, -12, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 11*x^14 - 12*x^13 + 83*x^12 - 96*x^11 + 219*x^10 - 294*x^9 + 1304*x^8 - 3846*x^7 + 1788*x^6 + 1980*x^5 - 7933*x^4 + 10752*x^3 + 26944*x^2 - 27624*x + 15856)
 
gp: K = bnfinit(x^16 - 11*x^14 - 12*x^13 + 83*x^12 - 96*x^11 + 219*x^10 - 294*x^9 + 1304*x^8 - 3846*x^7 + 1788*x^6 + 1980*x^5 - 7933*x^4 + 10752*x^3 + 26944*x^2 - 27624*x + 15856, 1)
 

Normalized defining polynomial

\( x^{16} - 11 x^{14} - 12 x^{13} + 83 x^{12} - 96 x^{11} + 219 x^{10} - 294 x^{9} + 1304 x^{8} - 3846 x^{7} + 1788 x^{6} + 1980 x^{5} - 7933 x^{4} + 10752 x^{3} + 26944 x^{2} - 27624 x + 15856 \)

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:  \(37319027463036582275390625=3^{8}\cdot 5^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.65$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(195=3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{195}(64,·)$, $\chi_{195}(1,·)$, $\chi_{195}(8,·)$, $\chi_{195}(77,·)$, $\chi_{195}(79,·)$, $\chi_{195}(83,·)$, $\chi_{195}(151,·)$, $\chi_{195}(92,·)$, $\chi_{195}(31,·)$, $\chi_{195}(34,·)$, $\chi_{195}(38,·)$, $\chi_{195}(109,·)$, $\chi_{195}(47,·)$, $\chi_{195}(181,·)$, $\chi_{195}(122,·)$, $\chi_{195}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{63684} a^{14} - \frac{505}{10614} a^{13} + \frac{853}{21228} a^{12} - \frac{335}{1769} a^{11} - \frac{10093}{63684} a^{10} - \frac{419}{5307} a^{9} - \frac{1}{36} a^{8} + \frac{1853}{10614} a^{7} - \frac{11083}{31842} a^{6} - \frac{1400}{5307} a^{5} + \frac{6419}{15921} a^{4} - \frac{4987}{10614} a^{3} + \frac{187}{7076} a^{2} + \frac{395}{5307} a + \frac{556}{15921}$, $\frac{1}{7008818593362299112509261208} a^{15} - \frac{120947892106551961507}{292034108056762463021219217} a^{14} - \frac{22448919001411702280103725}{2336272864454099704169753736} a^{13} + \frac{17334447458294512741563347}{292034108056762463021219217} a^{12} + \frac{550888533251635446371880383}{7008818593362299112509261208} a^{11} + \frac{38741901736329746392275911}{194689405371174975347479478} a^{10} + \frac{1564049126839298357650444771}{7008818593362299112509261208} a^{9} + \frac{225924324082050147052848869}{1168136432227049852084876868} a^{8} - \frac{2503676770598561501881091}{1752204648340574778127315302} a^{7} - \frac{2837143442639109013276261}{40280566628518960416719892} a^{6} - \frac{651897036777703073913556765}{1752204648340574778127315302} a^{5} - \frac{126679531560072621268936324}{292034108056762463021219217} a^{4} + \frac{218896202197566118968790811}{778757621484699901389917912} a^{3} - \frac{97824404237402261445430985}{584068216113524926042438434} a^{2} - \frac{809898097130579132293852469}{1752204648340574778127315302} a - \frac{25283649405903831370177795}{292034108056762463021219217}$

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

Class group and class number

$C_{2}\times C_{26}$, which has order $52$ (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:  \( 49732.9681714 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), 4.0.2471625.2, \(\Q(\sqrt{5}, \sqrt{13})\), 4.0.2471625.1, \(\Q(\zeta_{15})^+\), 4.4.190125.1, 4.0.2197.1, 4.0.54925.1, 8.0.6108930140625.4, 8.8.36147515625.1, 8.0.3016755625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed
13Data not computed