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Properties

Label 16.0.60404790201...000.32
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{8}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $129.22$
Ramified primes $2, 3, 5, 11$
Class number $1179648$ (GRH)
Class group $[16, 16, 48, 96]$ (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![37766834191, -10856135284, 13770521718, -3374395156, 2240623355, -454064500, 209524964, -34028752, 12191031, -1514672, 456524, -40348, 10903, -604, 154, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 154*x^14 - 604*x^13 + 10903*x^12 - 40348*x^11 + 456524*x^10 - 1514672*x^9 + 12191031*x^8 - 34028752*x^7 + 209524964*x^6 - 454064500*x^5 + 2240623355*x^4 - 3374395156*x^3 + 13770521718*x^2 - 10856135284*x + 37766834191)
 
gp: K = bnfinit(x^16 - 4*x^15 + 154*x^14 - 604*x^13 + 10903*x^12 - 40348*x^11 + 456524*x^10 - 1514672*x^9 + 12191031*x^8 - 34028752*x^7 + 209524964*x^6 - 454064500*x^5 + 2240623355*x^4 - 3374395156*x^3 + 13770521718*x^2 - 10856135284*x + 37766834191, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 154 x^{14} - 604 x^{13} + 10903 x^{12} - 40348 x^{11} + 456524 x^{10} - 1514672 x^{9} + 12191031 x^{8} - 34028752 x^{7} + 209524964 x^{6} - 454064500 x^{5} + 2240623355 x^{4} - 3374395156 x^{3} + 13770521718 x^{2} - 10856135284 x + 37766834191 \)

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:  \(6040479020157644046336000000000000=2^{44}\cdot 3^{8}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.22$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2640=2^{4}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2640}(1,·)$, $\chi_{2640}(197,·)$, $\chi_{2640}(2177,·)$, $\chi_{2640}(2573,·)$, $\chi_{2640}(2509,·)$, $\chi_{2640}(593,·)$, $\chi_{2640}(661,·)$, $\chi_{2640}(1849,·)$, $\chi_{2640}(857,·)$, $\chi_{2640}(1189,·)$, $\chi_{2640}(1253,·)$, $\chi_{2640}(529,·)$, $\chi_{2640}(1321,·)$, $\chi_{2640}(1517,·)$, $\chi_{2640}(1913,·)$, $\chi_{2640}(1981,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{217961} a^{12} + \frac{18832}{217961} a^{11} + \frac{21841}{217961} a^{10} - \frac{28953}{217961} a^{9} - \frac{24901}{217961} a^{8} + \frac{80449}{217961} a^{7} + \frac{19668}{217961} a^{6} + \frac{679}{217961} a^{5} - \frac{49230}{217961} a^{4} + \frac{53535}{217961} a^{3} + \frac{27488}{217961} a^{2} + \frac{86358}{217961} a - \frac{68485}{217961}$, $\frac{1}{217961} a^{13} + \frac{164}{217961} a^{11} - \frac{46258}{217961} a^{10} + \frac{97534}{217961} a^{9} - \frac{1161}{7031} a^{8} + \frac{51011}{217961} a^{7} - \frac{71358}{217961} a^{6} + \frac{23541}{217961} a^{5} - \frac{53199}{217961} a^{4} - \frac{74007}{217961} a^{3} + \frac{89717}{217961} a^{2} + \frac{62641}{217961} a + \frac{34283}{217961}$, $\frac{1}{314123946952260689} a^{14} + \frac{711627432525}{314123946952260689} a^{13} - \frac{569000454549}{314123946952260689} a^{12} - \frac{14884745470573155}{314123946952260689} a^{11} - \frac{38351036263884739}{314123946952260689} a^{10} + \frac{3928120169837595}{314123946952260689} a^{9} - \frac{48490987309341}{3976252493066591} a^{8} - \frac{147587961613179762}{314123946952260689} a^{7} + \frac{4184863075426808}{10133030546847119} a^{6} - \frac{115374457660821844}{314123946952260689} a^{5} + \frac{67848146429559557}{314123946952260689} a^{4} - \frac{56538393907663649}{314123946952260689} a^{3} + \frac{119763064843318619}{314123946952260689} a^{2} + \frac{91932587139811645}{314123946952260689} a - \frac{87905300807089187}{314123946952260689}$, $\frac{1}{12902742499192569553081880352402944441} a^{15} + \frac{11532464958598138489}{12902742499192569553081880352402944441} a^{14} + \frac{24032953219147175087860185398454}{12902742499192569553081880352402944441} a^{13} + \frac{700366886843911715906164984483}{12902742499192569553081880352402944441} a^{12} - \frac{27828954479862099630635593352146318}{163325854420159108266859244967125879} a^{11} + \frac{337398027978600372753264385763252699}{12902742499192569553081880352402944441} a^{10} - \frac{35459075155932490151663187154316789}{12902742499192569553081880352402944441} a^{9} + \frac{244527883573267077431644389951511808}{12902742499192569553081880352402944441} a^{8} + \frac{5567775114136193888370085888916131404}{12902742499192569553081880352402944441} a^{7} + \frac{2761663384815068358459919756475248583}{12902742499192569553081880352402944441} a^{6} - \frac{718954546766660737795102859239474722}{12902742499192569553081880352402944441} a^{5} - \frac{2747289923289687498891611810095887071}{12902742499192569553081880352402944441} a^{4} + \frac{136867603850744756532452777106855417}{416217499973953856551028398464611111} a^{3} + \frac{5777605861543859721922733797852262169}{12902742499192569553081880352402944441} a^{2} + \frac{3894900169528212149585563296566403594}{12902742499192569553081880352402944441} a + \frac{2314992893361887338177615329727912736}{12902742499192569553081880352402944441}$

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

Class group and class number

$C_{16}\times C_{16}\times C_{48}\times C_{96}$, which has order $1179648$ (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:  \( 12198.951274811623 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.51200.1, 4.0.8712000.5, 4.0.136125.2, 4.0.278784000.4, 4.0.278784000.2, 8.8.2621440000.1, 8.0.75898944000000.146, 8.0.77720518656000000.23

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$