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Properties

Label 16.16.4525912623...3889.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{14}\cdot 101^{14}$
Root discriminant $535.16$
Ramified primes $13, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61589352119, 285507154591, -58277636264, -185786927618, 46801723935, 27899474246, -7676264155, -1291290002, 396034194, 25863349, -9100211, -251544, 102277, 1240, -536, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 536*x^14 + 1240*x^13 + 102277*x^12 - 251544*x^11 - 9100211*x^10 + 25863349*x^9 + 396034194*x^8 - 1291290002*x^7 - 7676264155*x^6 + 27899474246*x^5 + 46801723935*x^4 - 185786927618*x^3 - 58277636264*x^2 + 285507154591*x + 61589352119)
 
gp: K = bnfinit(x^16 - 3*x^15 - 536*x^14 + 1240*x^13 + 102277*x^12 - 251544*x^11 - 9100211*x^10 + 25863349*x^9 + 396034194*x^8 - 1291290002*x^7 - 7676264155*x^6 + 27899474246*x^5 + 46801723935*x^4 - 185786927618*x^3 - 58277636264*x^2 + 285507154591*x + 61589352119, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 536 x^{14} + 1240 x^{13} + 102277 x^{12} - 251544 x^{11} - 9100211 x^{10} + 25863349 x^{9} + 396034194 x^{8} - 1291290002 x^{7} - 7676264155 x^{6} + 27899474246 x^{5} + 46801723935 x^{4} - 185786927618 x^{3} - 58277636264 x^{2} + 285507154591 x + 61589352119 \)

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:  \(45259126231720831581016789061101452813693889=13^{14}\cdot 101^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $535.16$
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:  $13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} + \frac{2}{17} a^{13} + \frac{8}{17} a^{12} + \frac{2}{17} a^{11} + \frac{3}{17} a^{10} - \frac{5}{17} a^{8} - \frac{7}{17} a^{5} - \frac{6}{17} a^{4} + \frac{4}{17} a^{3} + \frac{2}{17} a^{2} + \frac{3}{17} a - \frac{2}{17}$, $\frac{1}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{15} + \frac{9945481853522824614627992962822481762860810028201830464834230353734758245055623}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{14} + \frac{177080103354162562556942264036644611301644119138129791891186018370916991750012881}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{13} + \frac{48536344210040896202223401436731393297009104080587804754992998258836811443676823}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{12} - \frac{54810974581991208720958451880677997336676762776134512320092022119345267338634842}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{11} + \frac{102011800679009518884693568093250125094266351007724109610689554332724870491326604}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{10} + \frac{7278830388418417534537299606642975809920568625346533264998425036087797950727326}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{9} + \frac{9535586375421215596011927919046425216327243490769222856900339529882450967778233}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{8} + \frac{7266272546551819230725365495443313087598195752727076175547284687390908813718213}{26082145270273957818087558937541881667753266593124235695817526384961216316766337} a^{7} + \frac{4474062652520206060074456959335915018367433638565463696248850346343118039992696}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{6} - \frac{4385302647093813786254260097666949839878350345826365803551716337513667712725988}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{5} + \frac{149051080397098413388688575556480054333416516200689011791387211344743775323726052}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{4} - \frac{119949399769402748441178648750613989394511664331083242093590808234799518461687987}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{3} + \frac{84043970880074619499537905787467766195996160960387469335965759721033108043213047}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a^{2} - \frac{163359724734520466867553974694789362186998158474843350971671257582777422069527562}{443396469594657282907488501938211988351805532083112006828897948544340677385027729} a + \frac{186953092418745736695360379879692395517445751097192389870379805948218520388721547}{443396469594657282907488501938211988351805532083112006828897948544340677385027729}$

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:  \( 15798021542300000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{101}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{1313}) \), \(\Q(\sqrt{13}, \sqrt{101})\), 8.8.5123755016602262209.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$101$101.8.7.2$x^{8} - 404$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
101.8.7.2$x^{8} - 404$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$