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Properties

Label 16.0.72399383432...000.21
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 17^{14}$
Root discriminant $130.69$
Ramified primes $2, 3, 5, 17$
Class number $4737024$ (GRH)
Class group $[8, 24, 24672]$ (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![1057852414081, -62226612592, 238353759596, -12405931900, 24220970150, -1097221888, 1449580514, -55806478, 55896239, -1763952, 1422906, -34694, 23377, -394, 227, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 227*x^14 - 394*x^13 + 23377*x^12 - 34694*x^11 + 1422906*x^10 - 1763952*x^9 + 55896239*x^8 - 55806478*x^7 + 1449580514*x^6 - 1097221888*x^5 + 24220970150*x^4 - 12405931900*x^3 + 238353759596*x^2 - 62226612592*x + 1057852414081)
 
gp: K = bnfinit(x^16 - 2*x^15 + 227*x^14 - 394*x^13 + 23377*x^12 - 34694*x^11 + 1422906*x^10 - 1763952*x^9 + 55896239*x^8 - 55806478*x^7 + 1449580514*x^6 - 1097221888*x^5 + 24220970150*x^4 - 12405931900*x^3 + 238353759596*x^2 - 62226612592*x + 1057852414081, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 227 x^{14} - 394 x^{13} + 23377 x^{12} - 34694 x^{11} + 1422906 x^{10} - 1763952 x^{9} + 55896239 x^{8} - 55806478 x^{7} + 1449580514 x^{6} - 1097221888 x^{5} + 24220970150 x^{4} - 12405931900 x^{3} + 238353759596 x^{2} - 62226612592 x + 1057852414081 \)

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:  \(7239938343280505619539558400000000=2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.69$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2040=2^{3}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2040}(1,·)$, $\chi_{2040}(389,·)$, $\chi_{2040}(961,·)$, $\chi_{2040}(1801,·)$, $\chi_{2040}(749,·)$, $\chi_{2040}(1681,·)$, $\chi_{2040}(149,·)$, $\chi_{2040}(1441,·)$, $\chi_{2040}(869,·)$, $\chi_{2040}(361,·)$, $\chi_{2040}(1709,·)$, $\chi_{2040}(1589,·)$, $\chi_{2040}(841,·)$, $\chi_{2040}(121,·)$, $\chi_{2040}(509,·)$, $\chi_{2040}(1109,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{101} a^{14} + \frac{33}{101} a^{13} + \frac{15}{101} a^{12} - \frac{35}{101} a^{11} + \frac{31}{101} a^{10} - \frac{5}{101} a^{9} - \frac{13}{101} a^{8} + \frac{30}{101} a^{7} + \frac{46}{101} a^{6} - \frac{49}{101} a^{5} - \frac{26}{101} a^{4} + \frac{14}{101} a^{3} + \frac{40}{101} a^{2} + \frac{47}{101} a - \frac{28}{101}$, $\frac{1}{425722237968365333298592525109248082158359701464556719693} a^{15} + \frac{1244774192495044970277102151196831994265025832470400446}{425722237968365333298592525109248082158359701464556719693} a^{14} - \frac{35009090328654848240754401432490228267356530461038277423}{425722237968365333298592525109248082158359701464556719693} a^{13} + \frac{183760152496075612157746290519133391266783559989423867671}{425722237968365333298592525109248082158359701464556719693} a^{12} - \frac{72282030360670736399806941943878078514279145424893614325}{425722237968365333298592525109248082158359701464556719693} a^{11} - \frac{93419210285607085852652936431690729639597167581868487036}{425722237968365333298592525109248082158359701464556719693} a^{10} - \frac{87453160939790859940598961738255881602862656340272219965}{425722237968365333298592525109248082158359701464556719693} a^{9} - \frac{27745164161771350225897658602090096063594338625483253608}{425722237968365333298592525109248082158359701464556719693} a^{8} - \frac{84524763911458486003627904321445347371794086657555809270}{425722237968365333298592525109248082158359701464556719693} a^{7} + \frac{194635844654898470958319439460104375044534052889417306645}{425722237968365333298592525109248082158359701464556719693} a^{6} - \frac{172323445904169715537820966130302126395900640754348372798}{425722237968365333298592525109248082158359701464556719693} a^{5} + \frac{168930939268688217974525968658065967915469427064025698270}{425722237968365333298592525109248082158359701464556719693} a^{4} - \frac{55861259603920152941325641482264276186912956732161438841}{425722237968365333298592525109248082158359701464556719693} a^{3} - \frac{16322776171424895499459378578290387191604944903806225195}{425722237968365333298592525109248082158359701464556719693} a^{2} + \frac{112690761107224638499137675899119322000315226129359709533}{425722237968365333298592525109248082158359701464556719693} a - \frac{49684203175555038042815544342276722861477363268065498821}{425722237968365333298592525109248082158359701464556719693}$

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

Class group and class number

$C_{8}\times C_{24}\times C_{24672}$, which has order $4737024$ (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:  \( 3640.012213375973 \) (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

\(\Q(\sqrt{-510}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{17}, \sqrt{-30})\), 4.4.4913.1, 4.0.70747200.2, 8.0.5005166307840000.133, 8.0.85087827233280000.1, \(\Q(\zeta_{17})^+\)

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
3Data not computed
5Data not computed
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$