↑

Properties

Label 16.16.1868210871...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{24}\cdot 5^{12}\cdot 61^{6}\cdot 97^{4}$
Root discriminant $138.66$
Ramified primes $2, 5, 61, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T864

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3244799399, -6158326384, -1606936638, 2801999050, 1348808820, -400976426, -265146654, 19815150, 21657774, -209430, -893566, -9928, 19715, 280, -222, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 222*x^14 + 280*x^13 + 19715*x^12 - 9928*x^11 - 893566*x^10 - 209430*x^9 + 21657774*x^8 + 19815150*x^7 - 265146654*x^6 - 400976426*x^5 + 1348808820*x^4 + 2801999050*x^3 - 1606936638*x^2 - 6158326384*x - 3244799399)
 
gp: K = bnfinit(x^16 - 2*x^15 - 222*x^14 + 280*x^13 + 19715*x^12 - 9928*x^11 - 893566*x^10 - 209430*x^9 + 21657774*x^8 + 19815150*x^7 - 265146654*x^6 - 400976426*x^5 + 1348808820*x^4 + 2801999050*x^3 - 1606936638*x^2 - 6158326384*x - 3244799399, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 222 x^{14} + 280 x^{13} + 19715 x^{12} - 9928 x^{11} - 893566 x^{10} - 209430 x^{9} + 21657774 x^{8} + 19815150 x^{7} - 265146654 x^{6} - 400976426 x^{5} + 1348808820 x^{4} + 2801999050 x^{3} - 1606936638 x^{2} - 6158326384 x - 3244799399 \)

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:  \(18682108719227553550336000000000000=2^{24}\cdot 5^{12}\cdot 61^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $138.66$
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:  $2, 5, 61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{57} a^{14} + \frac{1}{57} a^{13} + \frac{2}{57} a^{12} + \frac{13}{57} a^{11} - \frac{20}{57} a^{10} - \frac{28}{57} a^{9} + \frac{1}{19} a^{8} - \frac{16}{57} a^{7} - \frac{5}{19} a^{6} + \frac{22}{57} a^{5} - \frac{1}{57} a^{4} + \frac{2}{19} a^{3} - \frac{5}{19} a^{2} - \frac{26}{57} a + \frac{17}{57}$, $\frac{1}{3938063342865992917030595044554014344791797192967498778897047} a^{15} + \frac{2674515061210766149232590084791322073625748335020273077675}{3938063342865992917030595044554014344791797192967498778897047} a^{14} - \frac{169584378375205430962038636192969456101324359215268407710740}{1312687780955330972343531681518004781597265730989166259632349} a^{13} - \frac{345780407725250753728810466567947070448172105013837388930325}{3938063342865992917030595044554014344791797192967498778897047} a^{12} - \frac{1852786252851225334746491119763525672099070052684988920114000}{3938063342865992917030595044554014344791797192967498778897047} a^{11} - \frac{1932239529321860643912075358152624089306242635659562686953154}{3938063342865992917030595044554014344791797192967498778897047} a^{10} - \frac{378391029300058845071996395253457998682324367764722312431385}{3938063342865992917030595044554014344791797192967498778897047} a^{9} + \frac{631610425094173011665741054979218536184450031624552297618164}{3938063342865992917030595044554014344791797192967498778897047} a^{8} + \frac{194523601893916362402725175382362825033204278612393662861611}{1312687780955330972343531681518004781597265730989166259632349} a^{7} + \frac{179857434157340376423562141396500011694042897560493037682392}{1312687780955330972343531681518004781597265730989166259632349} a^{6} - \frac{1094659406361822119127814319265475356822720057397831110140224}{3938063342865992917030595044554014344791797192967498778897047} a^{5} - \frac{731103035955130622541047619122778999442908803913500198662972}{3938063342865992917030595044554014344791797192967498778897047} a^{4} + \frac{211600770766117217991199057141389286400638745936397946821159}{1312687780955330972343531681518004781597265730989166259632349} a^{3} + \frac{169302746503203112811297640967986973679046477021626304497648}{1312687780955330972343531681518004781597265730989166259632349} a^{2} + \frac{1155289200892302412319839480307828153851836689063497248413359}{3938063342865992917030595044554014344791797192967498778897047} a + \frac{1453969975992512213054315316331745695337497842199770836033173}{3938063342865992917030595044554014344791797192967498778897047}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 923511289130 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T864:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n864
Character table for t16n864 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.14884000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$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}$
61Data not computed
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$