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Properties

Label 16.0.35847274805...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 59^{8}$
Root discriminant $25.68$
Ramified primes $5, 59$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50625, 3375, 225, -6735, -4274, 735, 932, 1081, 259, -156, 129, -98, -3, -7, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 7*x^13 - 3*x^12 - 98*x^11 + 129*x^10 - 156*x^9 + 259*x^8 + 1081*x^7 + 932*x^6 + 735*x^5 - 4274*x^4 - 6735*x^3 + 225*x^2 + 3375*x + 50625)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 - 7*x^13 - 3*x^12 - 98*x^11 + 129*x^10 - 156*x^9 + 259*x^8 + 1081*x^7 + 932*x^6 + 735*x^5 - 4274*x^4 - 6735*x^3 + 225*x^2 + 3375*x + 50625, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} - 7 x^{13} - 3 x^{12} - 98 x^{11} + 129 x^{10} - 156 x^{9} + 259 x^{8} + 1081 x^{7} + 932 x^{6} + 735 x^{5} - 4274 x^{4} - 6735 x^{3} + 225 x^{2} + 3375 x + 50625 \)

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:  \(35847274805742431640625=5^{12}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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:  $5, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{6} + \frac{3}{10} a$, $\frac{1}{70} a^{12} + \frac{1}{35} a^{11} + \frac{3}{70} a^{10} + \frac{1}{35} a^{9} + \frac{9}{35} a^{8} + \frac{3}{10} a^{7} + \frac{3}{35} a^{6} - \frac{31}{70} a^{5} + \frac{8}{35} a^{4} - \frac{13}{35} a^{3} - \frac{1}{70} a^{2} + \frac{2}{7} a - \frac{1}{14}$, $\frac{1}{2886346050} a^{13} + \frac{2120567}{577269210} a^{12} + \frac{10845949}{2886346050} a^{11} - \frac{137145739}{2886346050} a^{10} + \frac{8997769}{481057675} a^{9} + \frac{2975359}{6076518} a^{8} - \frac{359266857}{962115350} a^{7} + \frac{392901759}{962115350} a^{6} - \frac{151598741}{412335150} a^{5} + \frac{717508691}{1443173025} a^{4} - \frac{1364265289}{2886346050} a^{3} + \frac{19894309}{962115350} a^{2} + \frac{29695909}{115453842} a - \frac{6338349}{38484614}$, $\frac{1}{8659038150} a^{14} + \frac{1}{8659038150} a^{13} - \frac{43688081}{8659038150} a^{12} - \frac{2086963}{346361526} a^{11} - \frac{8067113}{288634605} a^{10} + \frac{724357909}{8659038150} a^{9} + \frac{810339893}{2886346050} a^{8} - \frac{70767133}{2886346050} a^{7} + \frac{284922077}{1731807630} a^{6} - \frac{54636242}{865903815} a^{5} + \frac{161547227}{455738850} a^{4} - \frac{1340832659}{2886346050} a^{3} + \frac{75086941}{1237005450} a^{2} + \frac{88043889}{192423070} a - \frac{8019218}{19242307}$, $\frac{1}{129885572250} a^{15} - \frac{1}{64942786125} a^{14} + \frac{2}{64942786125} a^{13} + \frac{344888884}{64942786125} a^{12} + \frac{522123187}{21647595375} a^{11} + \frac{1252199201}{64942786125} a^{10} - \frac{102833788}{3092513625} a^{9} + \frac{4566870874}{21647595375} a^{8} + \frac{8678222342}{64942786125} a^{7} - \frac{11338867147}{64942786125} a^{6} - \frac{137474084}{64942786125} a^{5} - \frac{16453909}{618502725} a^{4} - \frac{21159866362}{64942786125} a^{3} + \frac{224480363}{4329519075} a^{2} + \frac{10259920}{57726921} a + \frac{2581759}{5497802}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( -\frac{679}{41233515} a^{15} + \frac{1358}{41233515} a^{14} - \frac{4753}{82467030} a^{13} - \frac{679}{27489010} a^{12} + \frac{14669}{41233515} a^{11} + \frac{29197}{27489010} a^{10} - \frac{17654}{13744505} a^{9} + \frac{175861}{82467030} a^{8} + \frac{733999}{82467030} a^{7} - \frac{245447}{8246703} a^{6} + \frac{33271}{5497802} a^{5} - \frac{1451023}{41233515} a^{4} - \frac{304871}{5497802} a^{3} + \frac{10185}{5497802} a^{2} - \frac{14247076}{41233515} a + \frac{2291625}{5497802} \) (order $10$)
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:  \( 52746.8654076 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

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 $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-59}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-295}) \), \(\Q(\sqrt{5}, \sqrt{-59})\), 4.2.7375.1 x2, 4.0.435125.1 x2, 4.0.17405.1 x2, 4.2.1475.1 x2, 4.4.435125.1, \(\Q(\zeta_{5})\), 8.0.189333765625.2, 8.0.7573350625.1, 8.0.189333765625.1, 8.4.189333765625.1 x2, 8.0.54390625.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$59$59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$