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Properties

Label 16.4.12400595700...0409.6
Degree $16$
Signature $[4, 6]$
Discriminant $41^{14}\cdot 83^{6}$
Root discriminant $135.16$
Ramified primes $41, 83$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11447600, 413443500, 661786947, 169262104, -120605220, -8400012, 12450541, -7739418, 2453121, -681392, 148761, -30478, 5241, -492, 114, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 114*x^14 - 492*x^13 + 5241*x^12 - 30478*x^11 + 148761*x^10 - 681392*x^9 + 2453121*x^8 - 7739418*x^7 + 12450541*x^6 - 8400012*x^5 - 120605220*x^4 + 169262104*x^3 + 661786947*x^2 + 413443500*x - 11447600)
 
gp: K = bnfinit(x^16 - 2*x^15 + 114*x^14 - 492*x^13 + 5241*x^12 - 30478*x^11 + 148761*x^10 - 681392*x^9 + 2453121*x^8 - 7739418*x^7 + 12450541*x^6 - 8400012*x^5 - 120605220*x^4 + 169262104*x^3 + 661786947*x^2 + 413443500*x - 11447600, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 114 x^{14} - 492 x^{13} + 5241 x^{12} - 30478 x^{11} + 148761 x^{10} - 681392 x^{9} + 2453121 x^{8} - 7739418 x^{7} + 12450541 x^{6} - 8400012 x^{5} - 120605220 x^{4} + 169262104 x^{3} + 661786947 x^{2} + 413443500 x - 11447600 \)

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:  \(12400595700703321333205791744050409=41^{14}\cdot 83^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.16$
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:  $41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} + \frac{1}{8} a^{2}$, $\frac{1}{32} a^{12} - \frac{1}{16} a^{11} - \frac{1}{32} a^{10} - \frac{3}{32} a^{8} + \frac{1}{16} a^{7} - \frac{3}{32} a^{6} - \frac{1}{32} a^{4} - \frac{1}{16} a^{3} - \frac{15}{32} a^{2} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} - \frac{1}{16} a^{10} + \frac{1}{32} a^{9} - \frac{3}{32} a^{7} - \frac{1}{16} a^{6} - \frac{1}{32} a^{5} - \frac{3}{32} a^{3} - \frac{1}{16} a^{2}$, $\frac{1}{73310912} a^{14} + \frac{1102541}{73310912} a^{13} + \frac{111667}{18327728} a^{12} - \frac{587317}{73310912} a^{11} - \frac{2177527}{36655456} a^{10} - \frac{4257059}{73310912} a^{9} + \frac{4157537}{36655456} a^{8} - \frac{5004207}{73310912} a^{7} - \frac{3766841}{36655456} a^{6} - \frac{13449393}{73310912} a^{5} + \frac{1490171}{9163864} a^{4} + \frac{5772829}{73310912} a^{3} + \frac{22963015}{73310912} a^{2} - \frac{9068793}{18327728} a + \frac{611589}{4581932}$, $\frac{1}{78089845245309757978860125244661260077180138256205760} a^{15} + \frac{354386979057984049254948902026568554601284133}{78089845245309757978860125244661260077180138256205760} a^{14} - \frac{2692221645153586964653457486229813632599446564863}{2440307663915929936839378913895664377411879320506430} a^{13} + \frac{376317551482648624392763820678929974919739512190603}{78089845245309757978860125244661260077180138256205760} a^{12} + \frac{1557031931601538080599153886959700090524423811390923}{39044922622654878989430062622330630038590069128102880} a^{11} + \frac{1394794690320907802145232073570923017166921750426077}{78089845245309757978860125244661260077180138256205760} a^{10} - \frac{1844671465169152500070589812758231885517749627375397}{39044922622654878989430062622330630038590069128102880} a^{9} - \frac{8074080039265883134329888894074683784016846634715647}{78089845245309757978860125244661260077180138256205760} a^{8} - \frac{1205561730345490312572705592468489637549146716796187}{39044922622654878989430062622330630038590069128102880} a^{7} - \frac{774771424991801597433021825218195286267483408030913}{78089845245309757978860125244661260077180138256205760} a^{6} - \frac{1793810211582588953290739760596303914410560343294071}{19522461311327439494715031311165315019295034564051440} a^{5} + \frac{41785421637449979155131044005642704981042751030409}{2110536357981344810239462844450304326410274006924480} a^{4} + \frac{3287459724190504132178610564294122288080381554661191}{15617969049061951595772025048932252015436027651241152} a^{3} + \frac{3821685661409194968085347710883152057946543844698881}{19522461311327439494715031311165315019295034564051440} a^{2} + \frac{875242102712045024185221220087544202989374080155477}{4880615327831859873678757827791328754823758641012860} a + \frac{9753492908533477590858199644011095241737794702546}{244030766391592993683937891389566437741187932050643}$

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:  \( 231953121767 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.2.139523.2, 4.4.68921.1, 4.2.5720443.1, 8.4.1341662192766209.1 x2, 8.4.32723468116249.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$