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Properties

Label 16.0.26347107536...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{4}\cdot 5^{12}\cdot 59^{4}$
Root discriminant $68.99$
Ramified primes $2, 3, 5, 59$
Class number $96$ (GRH)
Class group $[2, 48]$ (GRH)
Galois group $C_2^4.C_2^4$ (as 16T459)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2052900, -463200, 1760200, -576800, 763220, -187680, 146320, -26560, 19376, -5160, 2612, -560, 154, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 + 154*x^12 - 560*x^11 + 2612*x^10 - 5160*x^9 + 19376*x^8 - 26560*x^7 + 146320*x^6 - 187680*x^5 + 763220*x^4 - 576800*x^3 + 1760200*x^2 - 463200*x + 2052900)
 
gp: K = bnfinit(x^16 + 8*x^14 + 154*x^12 - 560*x^11 + 2612*x^10 - 5160*x^9 + 19376*x^8 - 26560*x^7 + 146320*x^6 - 187680*x^5 + 763220*x^4 - 576800*x^3 + 1760200*x^2 - 463200*x + 2052900, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} + 154 x^{12} - 560 x^{11} + 2612 x^{10} - 5160 x^{9} + 19376 x^{8} - 26560 x^{7} + 146320 x^{6} - 187680 x^{5} + 763220 x^{4} - 576800 x^{3} + 1760200 x^{2} - 463200 x + 2052900 \)

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:  \(263471075369680896000000000000=2^{40}\cdot 3^{4}\cdot 5^{12}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.99$
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:  $2, 3, 5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{6} + \frac{3}{10} a^{4}$, $\frac{1}{60} a^{13} - \frac{1}{60} a^{12} - \frac{1}{30} a^{11} - \frac{2}{15} a^{10} + \frac{1}{15} a^{9} - \frac{7}{30} a^{8} - \frac{7}{15} a^{7} + \frac{2}{15} a^{6} + \frac{13}{30} a^{5} + \frac{7}{30} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{76260} a^{14} + \frac{509}{76260} a^{13} - \frac{199}{15252} a^{12} - \frac{587}{19065} a^{11} + \frac{260}{3813} a^{10} - \frac{1301}{19065} a^{9} + \frac{571}{3813} a^{8} - \frac{8623}{19065} a^{7} - \frac{493}{7626} a^{6} - \frac{683}{38130} a^{5} + \frac{4541}{38130} a^{4} + \frac{23}{123} a^{3} - \frac{332}{3813} a^{2} - \frac{605}{1271} a + \frac{575}{1271}$, $\frac{1}{2008134431298370476775330950576780} a^{15} - \frac{980734844049827897532963899}{167344535941530873064610912548065} a^{14} - \frac{3572694129622908607456380165707}{1004067215649185238387665475288390} a^{13} - \frac{5075396644623914644962705673043}{334689071883061746129221825096130} a^{12} - \frac{59940614953995559335814726573868}{502033607824592619193832737644195} a^{11} - \frac{65508287845596079372051349031716}{502033607824592619193832737644195} a^{10} - \frac{90949619342086717020053809132613}{1004067215649185238387665475288390} a^{9} + \frac{81579965076381913068618833834963}{334689071883061746129221825096130} a^{8} - \frac{279030658982476216779703985115569}{1004067215649185238387665475288390} a^{7} - \frac{25157650382828643228831600538162}{71719086832084659884833248234885} a^{6} - \frac{10520445782069751980728313469098}{502033607824592619193832737644195} a^{5} - \frac{19743639618386545554395266157024}{55781511980510291021536970849355} a^{4} + \frac{520278746011908374606601425185}{100406721564918523838766547528839} a^{3} + \frac{23427444646717547537840302121552}{100406721564918523838766547528839} a^{2} - \frac{4720376586905391747060853963360}{14343817366416931976966649646977} a + \frac{16652208914440366973260963896412}{33468907188306174612922182509613}$

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

Class group and class number

$C_{2}\times C_{48}$, which has order $96$ (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:  \( 2196592.73015 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^4$ (as 16T459):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^4.C_2^4$
Character table for $C_2^4.C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.118000.1, 4.0.472000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.3564544000000.19

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$