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Properties

Label 16.8.30341377703...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 5^{14}\cdot 41^{4}$
Root discriminant $69.60$
Ramified primes $2, 5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times D_4).C_2^3$ (as 16T315)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11325721, -14234616, -15956176, 40597764, -38194794, 21863772, -7407844, 407312, 603497, -181492, 13656, 2448, -644, 196, -36, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 36*x^14 + 196*x^13 - 644*x^12 + 2448*x^11 + 13656*x^10 - 181492*x^9 + 603497*x^8 + 407312*x^7 - 7407844*x^6 + 21863772*x^5 - 38194794*x^4 + 40597764*x^3 - 15956176*x^2 - 14234616*x + 11325721)
 
gp: K = bnfinit(x^16 - 4*x^15 - 36*x^14 + 196*x^13 - 644*x^12 + 2448*x^11 + 13656*x^10 - 181492*x^9 + 603497*x^8 + 407312*x^7 - 7407844*x^6 + 21863772*x^5 - 38194794*x^4 + 40597764*x^3 - 15956176*x^2 - 14234616*x + 11325721, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 36 x^{14} + 196 x^{13} - 644 x^{12} + 2448 x^{11} + 13656 x^{10} - 181492 x^{9} + 603497 x^{8} + 407312 x^{7} - 7407844 x^{6} + 21863772 x^{5} - 38194794 x^{4} + 40597764 x^{3} - 15956176 x^{2} - 14234616 x + 11325721 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(303413777032806400000000000000=2^{44}\cdot 5^{14}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.60$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{143} a^{14} + \frac{4}{13} a^{13} - \frac{36}{143} a^{12} + \frac{63}{143} a^{11} + \frac{48}{143} a^{10} - \frac{3}{13} a^{9} + \frac{71}{143} a^{8} + \frac{6}{143} a^{7} - \frac{49}{143} a^{6} + \frac{3}{11} a^{5} - \frac{46}{143} a^{4} + \frac{10}{143} a^{3} + \frac{40}{143} a^{2} + \frac{2}{11} a + \frac{4}{13}$, $\frac{1}{32233857917447709839022159966250532578164804227818200620147} a^{15} + \frac{42524521734790725442783550086721815738780377681648105566}{32233857917447709839022159966250532578164804227818200620147} a^{14} + \frac{3259032646665499076750036207362402698489309340550596487325}{32233857917447709839022159966250532578164804227818200620147} a^{13} - \frac{7697815821172347012217005442044185785193905873614282409321}{32233857917447709839022159966250532578164804227818200620147} a^{12} + \frac{11005869369330330909552544525268369229363143955822534510393}{32233857917447709839022159966250532578164804227818200620147} a^{11} - \frac{11765619217194216486831814440882001253464206776575768225877}{32233857917447709839022159966250532578164804227818200620147} a^{10} - \frac{10517224068953658464641951161568308168932298140679938656713}{32233857917447709839022159966250532578164804227818200620147} a^{9} + \frac{8026608987200002233329329354942178383410001846249711585357}{32233857917447709839022159966250532578164804227818200620147} a^{8} + \frac{1421175993271303401441215743586604268255507031203601998321}{32233857917447709839022159966250532578164804227818200620147} a^{7} + \frac{6208127268211468640509166872428290232196178820378984961405}{32233857917447709839022159966250532578164804227818200620147} a^{6} + \frac{5481570863647348723840150120593831172096349531868807133846}{32233857917447709839022159966250532578164804227818200620147} a^{5} + \frac{85544432994225520240511742232714717578776632311031867665}{32233857917447709839022159966250532578164804227818200620147} a^{4} - \frac{3147506397797327941216881787523105904242187489097915181032}{32233857917447709839022159966250532578164804227818200620147} a^{3} - \frac{11184282450811267886571387807572106405368491481813856096637}{32233857917447709839022159966250532578164804227818200620147} a^{2} + \frac{8596803748325589425710962589992582170651366301567870458693}{32233857917447709839022159966250532578164804227818200620147} a + \frac{563946604862085302035679676200786857767232789288348373155}{2930350719767973621729287269659139325287709475256200056377}$

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:  $11$
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:  \( 325563554.761 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4).C_2^3$ (as 16T315):

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 $(C_2\times D_4).C_2^3$
Character table for $(C_2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{20})^+\), 4.4.8000.1, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
41Data not computed