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Properties

Label 16.16.2751037671...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{12}\cdot 101^{12}$
Root discriminant $106.53$
Ramified primes $5, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![822625, 2731125, -1506750, -8288250, 2368075, 7870550, -3101025, -2333750, 1101440, 254535, -146355, -8750, 7381, 129, -149, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 149*x^14 + 129*x^13 + 7381*x^12 - 8750*x^11 - 146355*x^10 + 254535*x^9 + 1101440*x^8 - 2333750*x^7 - 3101025*x^6 + 7870550*x^5 + 2368075*x^4 - 8288250*x^3 - 1506750*x^2 + 2731125*x + 822625)
 
gp: K = bnfinit(x^16 - x^15 - 149*x^14 + 129*x^13 + 7381*x^12 - 8750*x^11 - 146355*x^10 + 254535*x^9 + 1101440*x^8 - 2333750*x^7 - 3101025*x^6 + 7870550*x^5 + 2368075*x^4 - 8288250*x^3 - 1506750*x^2 + 2731125*x + 822625, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 149 x^{14} + 129 x^{13} + 7381 x^{12} - 8750 x^{11} - 146355 x^{10} + 254535 x^{9} + 1101440 x^{8} - 2333750 x^{7} - 3101025 x^{6} + 7870550 x^{5} + 2368075 x^{4} - 8288250 x^{3} - 1506750 x^{2} + 2731125 x + 822625 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(275103767122062920083301025390625=5^{12}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.53$
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:  $5, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{4}$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{2}{5} a^{7} - \frac{1}{10} a^{6} - \frac{3}{10} a^{5} - \frac{1}{5} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{50} a^{11} - \frac{1}{50} a^{10} + \frac{1}{50} a^{9} + \frac{2}{25} a^{8} + \frac{21}{50} a^{7} - \frac{3}{10} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{10} a^{3} - \frac{1}{2} a$, $\frac{1}{150} a^{12} + \frac{1}{150} a^{11} - \frac{1}{150} a^{10} - \frac{7}{75} a^{9} + \frac{3}{50} a^{8} - \frac{1}{50} a^{7} + \frac{2}{5} a^{6} + \frac{1}{15} a^{5} + \frac{11}{30} a^{4} - \frac{4}{15} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{150} a^{13} + \frac{1}{150} a^{11} - \frac{1}{150} a^{10} + \frac{11}{150} a^{9} - \frac{1}{10} a^{8} + \frac{11}{25} a^{7} + \frac{4}{15} a^{6} + \frac{1}{5} a^{5} - \frac{13}{30} a^{4} - \frac{7}{15} a^{3} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{53250} a^{14} - \frac{61}{53250} a^{13} + \frac{61}{53250} a^{12} - \frac{401}{53250} a^{11} - \frac{74}{8875} a^{10} - \frac{4}{1065} a^{9} + \frac{112}{1775} a^{8} + \frac{1049}{10650} a^{7} + \frac{236}{5325} a^{6} + \frac{317}{1065} a^{5} - \frac{701}{2130} a^{4} + \frac{35}{426} a^{3} + \frac{68}{1065} a^{2} - \frac{61}{142} a - \frac{100}{213}$, $\frac{1}{54695641944540390245584206098385750} a^{15} - \frac{9284796595920287915966087031}{18231880648180130081861402032795250} a^{14} - \frac{5865653761539123206821751943587}{9115940324090065040930701016397625} a^{13} + \frac{53184904098380021659196712866536}{27347820972270195122792103049192875} a^{12} - \frac{3658768055096136340154879575414}{1439358998540536585410110686799625} a^{11} + \frac{1204909756908589187622979369583639}{27347820972270195122792103049192875} a^{10} - \frac{909269505649686388733384717436307}{10939128388908078049116841219677150} a^{9} + \frac{56385944050725145129931639956553}{575743599416214634164044274719850} a^{8} + \frac{34797704411150553186137226588503}{729275225927205203274456081311810} a^{7} + \frac{580806387433049001749270221092336}{1823188064818013008186140203279525} a^{6} - \frac{714593107946454849852512150253649}{2187825677781615609823368243935430} a^{5} - \frac{985933391051030699420005398042523}{2187825677781615609823368243935430} a^{4} - \frac{162192446132107319647796886119579}{1093912838890807804911684121967715} a^{3} - \frac{476957330663844332900537123749531}{1093912838890807804911684121967715} a^{2} + \frac{47032634934797841289965799524133}{437565135556323121964673648787086} a + \frac{186266271732375348890909114144131}{437565135556323121964673648787086}$

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:  $15$
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:  \( 124311294467 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{505}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{101})\), 4.4.51005.1 x2, 4.4.2525.1 x2, 4.4.128787625.1, 4.4.128787625.2, 8.8.65037750625.1, 8.8.16586252353140625.2, 8.8.16586252353140625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$