↑

Properties

Label 16.0.15478121650...0533.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{12}\cdot 157^{7}$
Root discriminant $137.04$
Ramified primes $37, 157$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1281

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1048347, 4064538, 5720470, 3416917, 1728007, 1246339, 317208, 133932, 91904, -6042, 15250, -2188, 1465, -192, 66, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 66*x^14 - 192*x^13 + 1465*x^12 - 2188*x^11 + 15250*x^10 - 6042*x^9 + 91904*x^8 + 133932*x^7 + 317208*x^6 + 1246339*x^5 + 1728007*x^4 + 3416917*x^3 + 5720470*x^2 + 4064538*x + 1048347)
 
gp: K = bnfinit(x^16 - 5*x^15 + 66*x^14 - 192*x^13 + 1465*x^12 - 2188*x^11 + 15250*x^10 - 6042*x^9 + 91904*x^8 + 133932*x^7 + 317208*x^6 + 1246339*x^5 + 1728007*x^4 + 3416917*x^3 + 5720470*x^2 + 4064538*x + 1048347, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 66 x^{14} - 192 x^{13} + 1465 x^{12} - 2188 x^{11} + 15250 x^{10} - 6042 x^{9} + 91904 x^{8} + 133932 x^{7} + 317208 x^{6} + 1246339 x^{5} + 1728007 x^{4} + 3416917 x^{3} + 5720470 x^{2} + 4064538 x + 1048347 \)

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:  \(15478121650083338314369033920290533=37^{12}\cdot 157^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $137.04$
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:  $37, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{333} a^{14} - \frac{16}{111} a^{13} - \frac{25}{333} a^{12} - \frac{128}{333} a^{11} - \frac{95}{333} a^{10} - \frac{32}{111} a^{9} + \frac{35}{111} a^{8} - \frac{4}{37} a^{7} + \frac{44}{333} a^{6} + \frac{163}{333} a^{5} - \frac{8}{37} a^{4} - \frac{14}{111} a^{3} - \frac{29}{333} a^{2} - \frac{49}{111} a + \frac{9}{37}$, $\frac{1}{17825544201148789123613898378810385865084313} a^{15} - \frac{836274807086143941660323055044661516403}{17825544201148789123613898378810385865084313} a^{14} + \frac{741859138190181517399605937390129149995438}{17825544201148789123613898378810385865084313} a^{13} - \frac{1543473172333928282161262402123559970166413}{17825544201148789123613898378810385865084313} a^{12} + \frac{533916031631629972496936011858111546015928}{1980616022349865458179322042090042873898257} a^{11} + \frac{497851680456481474511510964133171562648207}{17825544201148789123613898378810385865084313} a^{10} + \frac{902268701738452896139670434701186714159053}{5941848067049596374537966126270128621694771} a^{9} + \frac{362964511418864940369953830085400036614122}{5941848067049596374537966126270128621694771} a^{8} - \frac{8786069377306225445446294494437951671248418}{17825544201148789123613898378810385865084313} a^{7} + \frac{6976331932216195762389078353797973544068441}{17825544201148789123613898378810385865084313} a^{6} + \frac{4070664474687158378478293407681591178370185}{17825544201148789123613898378810385865084313} a^{5} - \frac{1258070796643908948264066785057653417877599}{5941848067049596374537966126270128621694771} a^{4} + \frac{6548530532503746041575342085501573789034508}{17825544201148789123613898378810385865084313} a^{3} - \frac{6364409111699195325829809023513503991240296}{17825544201148789123613898378810385865084313} a^{2} - \frac{2653875371350777328912973843320246949941203}{5941848067049596374537966126270128621694771} a - \frac{286273535529592942671229263329577862541465}{1980616022349865458179322042090042873898257}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 16595193587.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.50653.1, 8.0.63242590255441.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$37$37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.8.6.1$x^{8} - 1147 x^{4} + 855625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$157$157.2.1.2$x^{2} + 785$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.2$x^{2} + 785$$2$$1$$1$$C_2$$[\ ]_{2}$
157.4.2.1$x^{4} + 1727 x^{2} + 887364$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
157.4.3.4$x^{4} + 19625$$4$$1$$3$$C_4$$[\ ]_{4}$
157.4.0.1$x^{4} - x + 15$$1$$4$$0$$C_4$$[\ ]^{4}$