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Properties

Label 16.16.2584985751...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{14}\cdot 41^{6}$
Root discriminant $106.12$
Ramified primes $2, 3, 5, 41$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $(C_2\times Q_8).C_2^3$ (as 16T226)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-296375, 1823000, 3291250, -23793150, 2364300, 18195800, -4381410, -4645800, 1427926, 460454, -157868, -21158, 7455, 468, -148, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 148*x^14 + 468*x^13 + 7455*x^12 - 21158*x^11 - 157868*x^10 + 460454*x^9 + 1427926*x^8 - 4645800*x^7 - 4381410*x^6 + 18195800*x^5 + 2364300*x^4 - 23793150*x^3 + 3291250*x^2 + 1823000*x - 296375)
 
gp: K = bnfinit(x^16 - 4*x^15 - 148*x^14 + 468*x^13 + 7455*x^12 - 21158*x^11 - 157868*x^10 + 460454*x^9 + 1427926*x^8 - 4645800*x^7 - 4381410*x^6 + 18195800*x^5 + 2364300*x^4 - 23793150*x^3 + 3291250*x^2 + 1823000*x - 296375, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 148 x^{14} + 468 x^{13} + 7455 x^{12} - 21158 x^{11} - 157868 x^{10} + 460454 x^{9} + 1427926 x^{8} - 4645800 x^{7} - 4381410 x^{6} + 18195800 x^{5} + 2364300 x^{4} - 23793150 x^{3} + 3291250 x^{2} + 1823000 x - 296375 \)

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:  \(258498575149187174400000000000000=2^{24}\cdot 3^{12}\cdot 5^{14}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.12$
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:  $2, 3, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{8} - \frac{1}{5} a^{3}$, $\frac{1}{475} a^{14} - \frac{29}{475} a^{13} - \frac{13}{475} a^{12} + \frac{43}{475} a^{11} - \frac{2}{95} a^{10} + \frac{177}{475} a^{9} - \frac{23}{475} a^{8} - \frac{126}{475} a^{7} + \frac{4}{25} a^{6} - \frac{31}{95} a^{5} + \frac{32}{95} a^{4} + \frac{43}{95} a^{3} + \frac{8}{19} a^{2} - \frac{8}{19} a + \frac{1}{19}$, $\frac{1}{5264415472991624285225495703712401642324609275} a^{15} - \frac{445661038279161637909386139934525342067653}{1052883094598324857045099140742480328464921855} a^{14} + \frac{287047377720200578174874309890499739814521296}{5264415472991624285225495703712401642324609275} a^{13} - \frac{61751738399158687327493950851214288941877214}{5264415472991624285225495703712401642324609275} a^{12} + \frac{299085569599794464039340268738089766187118272}{5264415472991624285225495703712401642324609275} a^{11} + \frac{43857850585153485230686582874063840799860092}{5264415472991624285225495703712401642324609275} a^{10} - \frac{134026363552504906126420723996583600243026932}{1052883094598324857045099140742480328464921855} a^{9} - \frac{2299659451435382789471137413232337520120590073}{5264415472991624285225495703712401642324609275} a^{8} + \frac{1762377555240904321984015837073130820324139687}{5264415472991624285225495703712401642324609275} a^{7} + \frac{953243982426563871712788944956008150474880544}{5264415472991624285225495703712401642324609275} a^{6} + \frac{59751509877582755271004168243099723015052463}{210576618919664971409019828148496065692984371} a^{5} - \frac{49052521806152714161419502530011042545855077}{210576618919664971409019828148496065692984371} a^{4} - \frac{27098609600050657428281006066291275852994667}{210576618919664971409019828148496065692984371} a^{3} + \frac{84286874846753057314781609953957681178240286}{210576618919664971409019828148496065692984371} a^{2} - \frac{37763364054775684499966305540959888030951253}{210576618919664971409019828148496065692984371} a + \frac{23173252684917333452502552828760999461157232}{210576618919664971409019828148496065692984371}$

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

Class group and class number

$C_{4}$, 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:  $15$
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:  \( 37846609280.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times Q_8).C_2^3$ (as 16T226):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $(C_2\times Q_8).C_2^3$
Character table for $(C_2\times Q_8).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 4.4.16400.1, 4.4.738000.1, 8.8.544644000000.3

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$