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Properties

Label 16.0.31209693154...5424.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 127^{8}$
Root discriminant $165.35$
Ramified primes $2, 127$
Class number $23142400$ (GRH)
Class group $[2, 2, 2, 2, 2, 40, 18080]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1761875090942, -369689067024, 404754448936, -72093668176, 41025028336, -6174142000, 2397008128, -300845680, 88282889, -9006568, 2097020, -165704, 31318, -1736, 268, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 268*x^14 - 1736*x^13 + 31318*x^12 - 165704*x^11 + 2097020*x^10 - 9006568*x^9 + 88282889*x^8 - 300845680*x^7 + 2397008128*x^6 - 6174142000*x^5 + 41025028336*x^4 - 72093668176*x^3 + 404754448936*x^2 - 369689067024*x + 1761875090942)
 
gp: K = bnfinit(x^16 - 8*x^15 + 268*x^14 - 1736*x^13 + 31318*x^12 - 165704*x^11 + 2097020*x^10 - 9006568*x^9 + 88282889*x^8 - 300845680*x^7 + 2397008128*x^6 - 6174142000*x^5 + 41025028336*x^4 - 72093668176*x^3 + 404754448936*x^2 - 369689067024*x + 1761875090942, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 268 x^{14} - 1736 x^{13} + 31318 x^{12} - 165704 x^{11} + 2097020 x^{10} - 9006568 x^{9} + 88282889 x^{8} - 300845680 x^{7} + 2397008128 x^{6} - 6174142000 x^{5} + 41025028336 x^{4} - 72093668176 x^{3} + 404754448936 x^{2} - 369689067024 x + 1761875090942 \)

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:  \(312096931543105192096683948375015424=2^{62}\cdot 127^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $165.35$
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:  $2, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4064=2^{5}\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{4064}(1,·)$, $\chi_{4064}(2285,·)$, $\chi_{4064}(1017,·)$, $\chi_{4064}(3301,·)$, $\chi_{4064}(3809,·)$, $\chi_{4064}(3557,·)$, $\chi_{4064}(1777,·)$, $\chi_{4064}(2793,·)$, $\chi_{4064}(2541,·)$, $\chi_{4064}(253,·)$, $\chi_{4064}(2033,·)$, $\chi_{4064}(1269,·)$, $\chi_{4064}(3049,·)$, $\chi_{4064}(761,·)$, $\chi_{4064}(509,·)$, $\chi_{4064}(1525,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{31} a^{13} + \frac{9}{31} a^{12} - \frac{5}{31} a^{11} + \frac{6}{31} a^{10} - \frac{7}{31} a^{9} + \frac{11}{31} a^{8} - \frac{9}{31} a^{7} + \frac{5}{31} a^{6} + \frac{9}{31} a^{5} + \frac{11}{31} a^{4} + \frac{8}{31} a^{3} - \frac{7}{31} a^{2} - \frac{6}{31} a - \frac{13}{31}$, $\frac{1}{25402264508938058810557889} a^{14} - \frac{7}{25402264508938058810557889} a^{13} - \frac{1859712800597630566972985}{25402264508938058810557889} a^{12} + \frac{11158276803585783401838001}{25402264508938058810557889} a^{11} + \frac{3458380099506818726121148}{25402264508938058810557889} a^{10} + \frac{7435218014286519238668529}{25402264508938058810557889} a^{9} + \frac{871515194586054781969860}{25402264508938058810557889} a^{8} - \frac{1563117551433893571345984}{25402264508938058810557889} a^{7} + \frac{1265611736838201482292073}{25402264508938058810557889} a^{6} - \frac{931127178653007728023153}{25402264508938058810557889} a^{5} + \frac{3637030070341573679636163}{25402264508938058810557889} a^{4} + \frac{9318292477786691307983843}{25402264508938058810557889} a^{3} + \frac{7662285850962709818398793}{25402264508938058810557889} a^{2} + \frac{10351876300666297050549496}{25402264508938058810557889} a - \frac{5980863344086893751419902}{25402264508938058810557889}$, $\frac{1}{271218123796622501442033410127103} a^{15} + \frac{5338456}{271218123796622501442033410127103} a^{14} + \frac{2742910079125709439773297262424}{271218123796622501442033410127103} a^{13} - \frac{27756924420790340326796261014279}{271218123796622501442033410127103} a^{12} + \frac{103175632060409930335018997358425}{271218123796622501442033410127103} a^{11} - \frac{25261175830806316175397807249394}{271218123796622501442033410127103} a^{10} + \frac{85238586839073869435408762424886}{271218123796622501442033410127103} a^{9} - \frac{98749695621142527118075480729583}{271218123796622501442033410127103} a^{8} + \frac{80776501067351544498353479154876}{271218123796622501442033410127103} a^{7} + \frac{105336516611195194981907648737881}{271218123796622501442033410127103} a^{6} - \frac{125202997194842409085319805185375}{271218123796622501442033410127103} a^{5} + \frac{2359245927524588544040900648101}{8748971735374919401355916455713} a^{4} + \frac{100758252224929951063072513124232}{271218123796622501442033410127103} a^{3} + \frac{29847558615366908910393715237279}{271218123796622501442033410127103} a^{2} - \frac{118198678700158649949599713241123}{271218123796622501442033410127103} a + \frac{10758886495927872163679849408056}{271218123796622501442033410127103}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{40}\times C_{18080}$, which has order $23142400$ (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:  \( 15753.94986242651 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-127}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-254}) \), \(\Q(\sqrt{2}, \sqrt{-127})\), \(\Q(\zeta_{16})^+\), 4.0.33032192.5, 8.0.1091125708324864.35, \(\Q(\zeta_{32})^+\), 8.0.558656362662330368.31

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
$127$127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$