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Properties

Label 16.0.14390210149...481.11
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 89^{12}$
Root discriminant $242.59$
Ramified primes $17, 89$
Class number $19208$ (GRH)
Class group $[7, 14, 14, 14]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1373873435767232, -419116620051792, 59097374743052, -6943784680154, 1722809966457, 40074764600, 7734173792, 2092644532, -254586024, 6899572, -2708722, -36352, 18456, -1532, 20, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 20*x^14 - 1532*x^13 + 18456*x^12 - 36352*x^11 - 2708722*x^10 + 6899572*x^9 - 254586024*x^8 + 2092644532*x^7 + 7734173792*x^6 + 40074764600*x^5 + 1722809966457*x^4 - 6943784680154*x^3 + 59097374743052*x^2 - 419116620051792*x + 1373873435767232)
 
gp: K = bnfinit(x^16 - 2*x^15 + 20*x^14 - 1532*x^13 + 18456*x^12 - 36352*x^11 - 2708722*x^10 + 6899572*x^9 - 254586024*x^8 + 2092644532*x^7 + 7734173792*x^6 + 40074764600*x^5 + 1722809966457*x^4 - 6943784680154*x^3 + 59097374743052*x^2 - 419116620051792*x + 1373873435767232, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 20 x^{14} - 1532 x^{13} + 18456 x^{12} - 36352 x^{11} - 2708722 x^{10} + 6899572 x^{9} - 254586024 x^{8} + 2092644532 x^{7} + 7734173792 x^{6} + 40074764600 x^{5} + 1722809966457 x^{4} - 6943784680154 x^{3} + 59097374743052 x^{2} - 419116620051792 x + 1373873435767232 \)

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:  \(143902101499474565788603458614754288481=17^{12}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $242.59$
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:  $17, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{3}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{14} + \frac{3}{64} a^{12} - \frac{1}{32} a^{11} + \frac{1}{64} a^{10} - \frac{1}{8} a^{9} - \frac{3}{64} a^{8} + \frac{1}{32} a^{7} - \frac{1}{64} a^{6} + \frac{1}{16} a^{5} - \frac{15}{64} a^{4} + \frac{13}{32} a^{3} - \frac{3}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{91463341476263909110025533775773493288518894691346024367571258717147373044478815840518046501920477184} a^{15} + \frac{25060412838917027035288411574002090957181304715598908130042045437950280308979469177741590752221161}{91463341476263909110025533775773493288518894691346024367571258717147373044478815840518046501920477184} a^{14} - \frac{1117538371719042017018899679682035883176975330824904220953182936826818290456585806504760504681994249}{91463341476263909110025533775773493288518894691346024367571258717147373044478815840518046501920477184} a^{13} + \frac{378936589884687120644501674423743730995953855011196937291836463031871912492326018352155633568936129}{91463341476263909110025533775773493288518894691346024367571258717147373044478815840518046501920477184} a^{12} - \frac{4246365703054094286994069347236364377336560961841553752369567842421932129943430883748928847409755581}{91463341476263909110025533775773493288518894691346024367571258717147373044478815840518046501920477184} a^{11} + \frac{3898622706182360691279985513592465795644828161390866674600141798211130864256109022912660347670491265}{91463341476263909110025533775773493288518894691346024367571258717147373044478815840518046501920477184} a^{10} - \frac{11076775150517764790025971358292998696015395956371216816743528451545153076509097539386477681752816007}{91463341476263909110025533775773493288518894691346024367571258717147373044478815840518046501920477184} a^{9} - \frac{2488105455895765923099405674674442763351436707931552008909852791555216810382506963926422303999881849}{91463341476263909110025533775773493288518894691346024367571258717147373044478815840518046501920477184} a^{8} - \frac{9636786897537547843707055120453717330644619233530550814693326335552788106902777808527469186549002939}{91463341476263909110025533775773493288518894691346024367571258717147373044478815840518046501920477184} a^{7} - \frac{14636448470699780781882059013190778876068576106996525955122856747924688577549494695628861698562214901}{91463341476263909110025533775773493288518894691346024367571258717147373044478815840518046501920477184} a^{6} + \frac{19063509457092029450656298938558239755330858372679094831664122987878600716381807911485230893203392889}{91463341476263909110025533775773493288518894691346024367571258717147373044478815840518046501920477184} a^{5} + \frac{4623438851471221138225997716723223345523621700009118835267125030734218828841607519816569378061646923}{91463341476263909110025533775773493288518894691346024367571258717147373044478815840518046501920477184} a^{4} - \frac{7714271122200120146653299874117583593779024079775533177300082920300521537031556430606427810038564951}{45731670738131954555012766887886746644259447345673012183785629358573686522239407920259023250960238592} a^{3} + \frac{2124995326788977551344290779815464917498101550829904816238170951384555834571588107458027287222867803}{22865835369065977277506383443943373322129723672836506091892814679286843261119703960129511625480119296} a^{2} - \frac{482203277501889463393657928798207469750979566667877456784205490917564551391146008054016926758922429}{5716458842266494319376595860985843330532430918209126522973203669821710815279925990032377906370029824} a + \frac{349237509733390811431760458881436697414672103185596598284571538737785797044279059347465089989769131}{1429114710566623579844148965246460832633107729552281630743300917455427703819981497508094476592507456}$

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

Class group and class number

$C_{7}\times C_{14}\times C_{14}\times C_{14}$, which has order $19208$ (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:  \( 715570753.662 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

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 $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{1513}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{17}, \sqrt{89})\), 4.0.25721.1 x2, 4.0.134657.1 x2, 4.4.3463512697.2, 4.4.3463512697.1, 8.0.5240294710561.2, 8.8.11995920202280213809.1, 8.0.11995920202280213809.4

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
$89$89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$