Properties

Label 16.16.462...081.1
Degree $16$
Signature $[16, 0]$
Discriminant $4.627\times 10^{44}$
Root discriminant $618.85$
Ramified primes $29, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 521*x^14 - 352*x^13 + 97262*x^12 + 111457*x^11 - 8578319*x^10 - 7737475*x^9 + 391096418*x^8 + 106474606*x^7 - 9220425289*x^6 + 3780858515*x^5 + 102878217925*x^4 - 87419452512*x^3 - 457661368284*x^2 + 433370758320*x + 437343265328)
 
gp: K = bnfinit(x^16 - x^15 - 521*x^14 - 352*x^13 + 97262*x^12 + 111457*x^11 - 8578319*x^10 - 7737475*x^9 + 391096418*x^8 + 106474606*x^7 - 9220425289*x^6 + 3780858515*x^5 + 102878217925*x^4 - 87419452512*x^3 - 457661368284*x^2 + 433370758320*x + 437343265328, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![437343265328, 433370758320, -457661368284, -87419452512, 102878217925, 3780858515, -9220425289, 106474606, 391096418, -7737475, -8578319, 111457, 97262, -352, -521, -1, 1]);
 

\( x^{16} - x^{15} - 521 x^{14} - 352 x^{13} + 97262 x^{12} + 111457 x^{11} - 8578319 x^{10} - 7737475 x^{9} + 391096418 x^{8} + 106474606 x^{7} - 9220425289 x^{6} + 3780858515 x^{5} + 102878217925 x^{4} - 87419452512 x^{3} - 457661368284 x^{2} + 433370758320 x + 437343265328 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[16, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(462732306245995722656474121747020143758680081\)\(\medspace = 29^{14}\cdot 41^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $618.85$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $29, 41$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{236} a^{13} - \frac{13}{118} a^{12} - \frac{7}{118} a^{11} + \frac{25}{118} a^{10} + \frac{23}{118} a^{9} - \frac{99}{236} a^{8} + \frac{13}{118} a^{7} - \frac{14}{59} a^{6} + \frac{29}{118} a^{5} + \frac{29}{59} a^{4} - \frac{51}{236} a^{3} + \frac{9}{118} a^{2} + \frac{17}{59} a + \frac{10}{59}$, $\frac{1}{65608} a^{14} - \frac{135}{65608} a^{13} + \frac{1109}{65608} a^{12} - \frac{4817}{32804} a^{11} + \frac{10455}{32804} a^{10} + \frac{29461}{65608} a^{9} - \frac{12901}{65608} a^{8} - \frac{12153}{65608} a^{7} + \frac{6821}{16402} a^{6} - \frac{7587}{32804} a^{5} - \frac{32165}{65608} a^{4} + \frac{23985}{65608} a^{3} + \frac{31323}{65608} a^{2} + \frac{16197}{32804} a + \frac{6875}{16402}$, $\frac{1}{321035756176800321337459069415982798473937207108038960888283649879421008} a^{15} - \frac{1158644171214648419675466290975699187954007284325791446679804897381}{321035756176800321337459069415982798473937207108038960888283649879421008} a^{14} + \frac{303754550184691814488120744005105730464551706084901105203036431427775}{321035756176800321337459069415982798473937207108038960888283649879421008} a^{13} - \frac{4072922036129412747748348592672165821048977068694752857566825193421427}{80258939044200080334364767353995699618484301777009740222070912469855252} a^{12} - \frac{31558603806258229750859464989420502702240631702357836820808995351461775}{160517878088400160668729534707991399236968603554019480444141824939710504} a^{11} - \frac{30948592906570676610186101270999404768131154622651171571927652212017303}{321035756176800321337459069415982798473937207108038960888283649879421008} a^{10} + \frac{82253407873718542161830055927151628084427914483729753542819803818574253}{321035756176800321337459069415982798473937207108038960888283649879421008} a^{9} + \frac{30861789342179779599528774045615254731335103033206979530059168854119093}{321035756176800321337459069415982798473937207108038960888283649879421008} a^{8} - \frac{37647327168511590353257372402839670282555682365560762658993833034365477}{160517878088400160668729534707991399236968603554019480444141824939710504} a^{7} + \frac{32620331139310392713501630099016848878443272748526294213603071702496145}{160517878088400160668729534707991399236968603554019480444141824939710504} a^{6} - \frac{43249850811297512642457549120298053521349851051632546515375263540426193}{321035756176800321337459069415982798473937207108038960888283649879421008} a^{5} + \frac{29113854011123939448783532533951335619716432855850426609209476541195303}{321035756176800321337459069415982798473937207108038960888283649879421008} a^{4} - \frac{132384494305102130014089595765244255971788289456627326984117563391754683}{321035756176800321337459069415982798473937207108038960888283649879421008} a^{3} - \frac{37309707977724838863602661744213478950854223584924055721068239263787285}{80258939044200080334364767353995699618484301777009740222070912469855252} a^{2} + \frac{3655808263463726653623206245515932377771774414467890254278452306437114}{20064734761050020083591191838498924904621075444252435055517728117463813} a + \frac{2562996301598481816209831867085095907555422439814516616982538910678367}{40129469522100040167182383676997849809242150888504870111035456234927626}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 90185580600800000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{16}\cdot(2\pi)^{0}\cdot 90185580600800000 \cdot 2}{2\sqrt{462732306245995722656474121747020143758680081}}\approx 0.274759125524566$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.57962561.1, 8.8.115844383968839978801.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
29Data not computed
41Data not computed