↑

Properties

Label 16.0.19997735500...6816.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 101^{8}$
Root discriminant $139.26$
Ramified primes $2, 3, 101$
Class number $6504120$ (GRH)
Class group $[3, 2168040]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![363345076222, -1678274856, 87721316756, -69573768, 9632982138, 4689888, 631526924, 320448, 27168519, 4536, 789956, -24, 15254, 0, 180, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 180*x^14 + 15254*x^12 - 24*x^11 + 789956*x^10 + 4536*x^9 + 27168519*x^8 + 320448*x^7 + 631526924*x^6 + 4689888*x^5 + 9632982138*x^4 - 69573768*x^3 + 87721316756*x^2 - 1678274856*x + 363345076222)
 
gp: K = bnfinit(x^16 + 180*x^14 + 15254*x^12 - 24*x^11 + 789956*x^10 + 4536*x^9 + 27168519*x^8 + 320448*x^7 + 631526924*x^6 + 4689888*x^5 + 9632982138*x^4 - 69573768*x^3 + 87721316756*x^2 - 1678274856*x + 363345076222, 1)
 

Normalized defining polynomial

\( x^{16} + 180 x^{14} + 15254 x^{12} - 24 x^{11} + 789956 x^{10} + 4536 x^{9} + 27168519 x^{8} + 320448 x^{7} + 631526924 x^{6} + 4689888 x^{5} + 9632982138 x^{4} - 69573768 x^{3} + 87721316756 x^{2} - 1678274856 x + 363345076222 \)

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:  \(19997735500105260525348149910306816=2^{48}\cdot 3^{8}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $139.26$
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:  $2, 3, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4848=2^{4}\cdot 3\cdot 101\)
Dirichlet character group:    $\lbrace$$\chi_{4848}(1,·)$, $\chi_{4848}(2627,·)$, $\chi_{4848}(1415,·)$, $\chi_{4848}(2827,·)$, $\chi_{4848}(1615,·)$, $\chi_{4848}(4241,·)$, $\chi_{4848}(403,·)$, $\chi_{4848}(3029,·)$, $\chi_{4848}(203,·)$, $\chi_{4848}(1817,·)$, $\chi_{4848}(605,·)$, $\chi_{4848}(4039,·)$, $\chi_{4848}(3637,·)$, $\chi_{4848}(2425,·)$, $\chi_{4848}(1213,·)$, $\chi_{4848}(3839,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5704589327} a^{14} - \frac{91033435}{5704589327} a^{13} - \frac{1638552044}{5704589327} a^{12} + \frac{2790957540}{5704589327} a^{11} + \frac{273113556}{5704589327} a^{10} + \frac{97996122}{5704589327} a^{9} + \frac{1260767226}{5704589327} a^{8} + \frac{2342368966}{5704589327} a^{7} - \frac{2614655181}{5704589327} a^{6} - \frac{184665889}{5704589327} a^{5} - \frac{963084587}{5704589327} a^{4} + \frac{1697970214}{5704589327} a^{3} + \frac{1504175107}{5704589327} a^{2} - \frac{747741675}{5704589327} a - \frac{387449402}{5704589327}$, $\frac{1}{1026028165756252775486476074193191798458826402479} a^{15} + \frac{68682752567712861402773966239999841224}{1026028165756252775486476074193191798458826402479} a^{14} + \frac{333978247371588753306399082526025511657675121566}{1026028165756252775486476074193191798458826402479} a^{13} - \frac{391338949779946418677077880447787861716652918288}{1026028165756252775486476074193191798458826402479} a^{12} + \frac{247442599070710882036084734036220032011554978864}{1026028165756252775486476074193191798458826402479} a^{11} - \frac{9113035203496082803931348786681947260485759165}{21830386505452186712478214344535995711889923457} a^{10} - \frac{323313232675108048828287623383194296725799921010}{1026028165756252775486476074193191798458826402479} a^{9} - \frac{101317505644090800385659618678628607453564469350}{1026028165756252775486476074193191798458826402479} a^{8} + \frac{16874191080945180058160198449630102710498734171}{1026028165756252775486476074193191798458826402479} a^{7} - \frac{223064480656100917624014722581536195853493578427}{1026028165756252775486476074193191798458826402479} a^{6} - \frac{482617840450330003904126945832203397473305247006}{1026028165756252775486476074193191798458826402479} a^{5} - \frac{484851155975382097509114484689546875906440215485}{1026028165756252775486476074193191798458826402479} a^{4} - \frac{507060352891441509272240052089472328364081899475}{1026028165756252775486476074193191798458826402479} a^{3} + \frac{67268899730145273433379289455432899271447400593}{1026028165756252775486476074193191798458826402479} a^{2} + \frac{37259477828818201630352277448593619564643444434}{1026028165756252775486476074193191798458826402479} a - \frac{508892181498785537825088292656237112529389592042}{1026028165756252775486476074193191798458826402479}$

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

Class group and class number

$C_{3}\times C_{2168040}$, which has order $6504120$ (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:  \( 11964.310642723332 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-101}) \), \(\Q(\sqrt{-303}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-202}) \), \(\Q(\sqrt{-606}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}, \sqrt{-101})\), \(\Q(\sqrt{2}, \sqrt{-101})\), \(\Q(\sqrt{6}, \sqrt{-101})\), \(\Q(\sqrt{2}, \sqrt{-303})\), \(\Q(\sqrt{6}, \sqrt{-202})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-202})\), 4.0.188024832.2, 4.4.18432.1, \(\Q(\zeta_{16})^+\), 4.0.20891648.2, 8.0.552395897634816.13, 8.0.141413349794512896.5, 8.0.1745843824623616.9, 8.0.35353337448628224.54, 8.0.35353337448628224.29, 8.0.141413349794512896.6, \(\Q(\zeta_{48})^+\)

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$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}$
$101$101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$