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Properties

Label 16.16.3638714695...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{40}\cdot 3^{6}\cdot 5^{6}\cdot 13^{8}\cdot 1249^{2}\cdot 1511^{2}$
Root discriminant $342.81$
Ramified primes $2, 3, 5, 13, 1249, 1511$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9414762766137, -13332861193500, -3349786075272, 3910405446324, 544143437772, -367215551292, -54499332552, 13272467508, 2442903326, -137669268, -37641976, 362012, 258716, 1004, -824, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 824*x^14 + 1004*x^13 + 258716*x^12 + 362012*x^11 - 37641976*x^10 - 137669268*x^9 + 2442903326*x^8 + 13272467508*x^7 - 54499332552*x^6 - 367215551292*x^5 + 544143437772*x^4 + 3910405446324*x^3 - 3349786075272*x^2 - 13332861193500*x + 9414762766137)
 
gp: K = bnfinit(x^16 - 4*x^15 - 824*x^14 + 1004*x^13 + 258716*x^12 + 362012*x^11 - 37641976*x^10 - 137669268*x^9 + 2442903326*x^8 + 13272467508*x^7 - 54499332552*x^6 - 367215551292*x^5 + 544143437772*x^4 + 3910405446324*x^3 - 3349786075272*x^2 - 13332861193500*x + 9414762766137, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 824 x^{14} + 1004 x^{13} + 258716 x^{12} + 362012 x^{11} - 37641976 x^{10} - 137669268 x^{9} + 2442903326 x^{8} + 13272467508 x^{7} - 54499332552 x^{6} - 367215551292 x^{5} + 544143437772 x^{4} + 3910405446324 x^{3} - 3349786075272 x^{2} - 13332861193500 x + 9414762766137 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(36387146957833308423870594234187776000000=2^{40}\cdot 3^{6}\cdot 5^{6}\cdot 13^{8}\cdot 1249^{2}\cdot 1511^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $342.81$
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:  $2, 3, 5, 13, 1249, 1511$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} + \frac{1}{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} + \frac{1}{8} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{4} - \frac{1}{8} a^{2} + \frac{1}{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{16} a^{3} + \frac{3}{16} a^{2} - \frac{1}{16} a + \frac{3}{16}$, $\frac{1}{48} a^{12} - \frac{1}{48} a^{11} + \frac{1}{48} a^{10} - \frac{1}{48} a^{9} + \frac{1}{24} a^{8} + \frac{1}{24} a^{7} + \frac{1}{24} a^{6} - \frac{1}{8} a^{5} + \frac{5}{48} a^{4} - \frac{3}{16} a^{3} + \frac{7}{16} a^{2} + \frac{5}{16} a$, $\frac{1}{1104} a^{13} + \frac{11}{1104} a^{12} + \frac{5}{552} a^{11} - \frac{1}{276} a^{10} - \frac{13}{1104} a^{9} + \frac{41}{1104} a^{8} - \frac{11}{138} a^{7} - \frac{5}{46} a^{6} + \frac{131}{1104} a^{5} - \frac{73}{368} a^{4} - \frac{7}{184} a^{3} + \frac{35}{92} a^{2} - \frac{101}{368} a + \frac{21}{368}$, $\frac{1}{2208} a^{14} - \frac{19}{2208} a^{12} - \frac{17}{552} a^{11} - \frac{5}{736} a^{10} + \frac{1}{24} a^{9} - \frac{79}{2208} a^{8} - \frac{3}{92} a^{7} - \frac{7}{736} a^{6} - \frac{35}{276} a^{5} + \frac{343}{2208} a^{4} - \frac{7}{184} a^{3} - \frac{31}{736} a^{2} + \frac{53}{184} a - \frac{231}{736}$, $\frac{1}{170540134734538709797500428634863536014374774958142188357731677171841944992} a^{15} - \frac{7801261266001442200238034370268798509469519840092912027227714599106217}{56846711578179569932500142878287845338124924986047396119243892390613981664} a^{14} - \frac{31925909543462524402957638136218972899139307196501095696641757195056035}{170540134734538709797500428634863536014374774958142188357731677171841944992} a^{13} - \frac{925650340274357156696058574837183629370112438954025616151756861232692725}{170540134734538709797500428634863536014374774958142188357731677171841944992} a^{12} - \frac{4661261052067093457520301162946030911809311356594789521203281197205286119}{170540134734538709797500428634863536014374774958142188357731677171841944992} a^{11} - \frac{2384441648929247379995306446376003772826801363970988184260382881428469381}{56846711578179569932500142878287845338124924986047396119243892390613981664} a^{10} - \frac{64830142442857750078427425551558896012845680299028937734672777435305047}{2471596155573024779673919255577732406005431521132495483445386625678868768} a^{9} - \frac{1401247566863112601828206924186763208266567969676780047377786124666565189}{170540134734538709797500428634863536014374774958142188357731677171841944992} a^{8} + \frac{3902666319968069208155364766983992868172899918566679264763611569014541275}{170540134734538709797500428634863536014374774958142188357731677171841944992} a^{7} - \frac{222595511175636929236263461003926396701812627366331140848315031130993625}{170540134734538709797500428634863536014374774958142188357731677171841944992} a^{6} + \frac{41144559155933743988480239671242992612761573006019367891884608546130775663}{170540134734538709797500428634863536014374774958142188357731677171841944992} a^{5} - \frac{7057867514974744127338158965892760100954784405394127386372976000338007797}{56846711578179569932500142878287845338124924986047396119243892390613981664} a^{4} - \frac{7787000786386601321755033339895801452304986407023793135723391214570803831}{56846711578179569932500142878287845338124924986047396119243892390613981664} a^{3} + \frac{26089641634945662231383128802667424257991770731009399003187530511573563377}{56846711578179569932500142878287845338124924986047396119243892390613981664} a^{2} + \frac{16814383931825062663210161249133901697519906937944868329895644431307734781}{56846711578179569932500142878287845338124924986047396119243892390613981664} a - \frac{25697548932286380074102458064727974216327125943071348859157321255132302653}{56846711578179569932500142878287845338124924986047396119243892390613981664}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 743643457753000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 133 conjugacy class representatives for t16n1547 are not computed
Character table for t16n1547 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{2}, \sqrt{13})\), 8.8.421149081600.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13Data not computed
1249Data not computed
1511Data not computed