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Properties

Label 16.4.40745431714...4569.7
Degree $16$
Signature $[4, 6]$
Discriminant $13^{12}\cdot 53^{10}$
Root discriminant $81.87$
Ramified primes $13, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-469741, 1148831, -714532, -614407, 953906, -123952, -307004, 114933, 16465, -17437, 5717, 444, -415, 136, 3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 3*x^14 + 136*x^13 - 415*x^12 + 444*x^11 + 5717*x^10 - 17437*x^9 + 16465*x^8 + 114933*x^7 - 307004*x^6 - 123952*x^5 + 953906*x^4 - 614407*x^3 - 714532*x^2 + 1148831*x - 469741)
 
gp: K = bnfinit(x^16 - 4*x^15 + 3*x^14 + 136*x^13 - 415*x^12 + 444*x^11 + 5717*x^10 - 17437*x^9 + 16465*x^8 + 114933*x^7 - 307004*x^6 - 123952*x^5 + 953906*x^4 - 614407*x^3 - 714532*x^2 + 1148831*x - 469741, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 3 x^{14} + 136 x^{13} - 415 x^{12} + 444 x^{11} + 5717 x^{10} - 17437 x^{9} + 16465 x^{8} + 114933 x^{7} - 307004 x^{6} - 123952 x^{5} + 953906 x^{4} - 614407 x^{3} - 714532 x^{2} + 1148831 x - 469741 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4074543171431096342049568754569=13^{12}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.87$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{56} a^{14} - \frac{1}{7} a^{13} + \frac{3}{14} a^{12} - \frac{1}{7} a^{11} + \frac{13}{56} a^{10} + \frac{2}{7} a^{9} - \frac{11}{28} a^{8} - \frac{3}{8} a^{7} - \frac{25}{56} a^{6} - \frac{3}{14} a^{5} - \frac{5}{56} a^{4} - \frac{1}{7} a^{3} - \frac{19}{56} a^{2} + \frac{5}{56} a - \frac{3}{56}$, $\frac{1}{89323566585868090746870768402635418864} a^{15} + \frac{78275165915198051749701815782719629}{29774522195289363582290256134211806288} a^{14} + \frac{29681950811992014900986325592088935}{1063375792688905842224652004793278796} a^{13} - \frac{3466270983625169100720376504642429871}{22330891646467022686717692100658854716} a^{12} + \frac{634720439583027241325406901598647255}{29774522195289363582290256134211806288} a^{11} + \frac{7318442881384663323101529600513502481}{29774522195289363582290256134211806288} a^{10} + \frac{1947127111699773381260723403076554791}{4060162117539458670312307654665246312} a^{9} + \frac{27678454285093587008979238592349925345}{89323566585868090746870768402635418864} a^{8} + \frac{70113589076262053566964637991959205}{2030081058769729335156153827332623156} a^{7} + \frac{5596034761858081724856510383186897525}{89323566585868090746870768402635418864} a^{6} + \frac{8499411652512993364120751452039097501}{29774522195289363582290256134211806288} a^{5} + \frac{25497181852678745358396488825962822901}{89323566585868090746870768402635418864} a^{4} - \frac{27839023687862503392782965758875605907}{89323566585868090746870768402635418864} a^{3} - \frac{1543243193951479201660887593643925161}{3721815274411170447786282016776475786} a^{2} + \frac{193415400401395516792717593876058091}{5582722911616755671679423025164713679} a + \frac{177246863362205309361427334940527987}{790474040582903457936909454890578928}$

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:  $9$
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:  \( 759777887.761 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.8957.1, 8.4.38085844705129.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$