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Properties

Label 16.0.72699242014...7296.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 89^{8}$
Root discriminant $130.72$
Ramified primes $2, 3, 89$
Class number $1297920$ (GRH)
Class group $[4, 52, 6240]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![145330457809, -926997912, 38637575756, -43671120, 4685677056, 3145200, 340542452, 247008, 16317753, 4032, 531680, -24, 11594, 0, 156, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 156*x^14 + 11594*x^12 - 24*x^11 + 531680*x^10 + 4032*x^9 + 16317753*x^8 + 247008*x^7 + 340542452*x^6 + 3145200*x^5 + 4685677056*x^4 - 43671120*x^3 + 38637575756*x^2 - 926997912*x + 145330457809)
 
gp: K = bnfinit(x^16 + 156*x^14 + 11594*x^12 - 24*x^11 + 531680*x^10 + 4032*x^9 + 16317753*x^8 + 247008*x^7 + 340542452*x^6 + 3145200*x^5 + 4685677056*x^4 - 43671120*x^3 + 38637575756*x^2 - 926997912*x + 145330457809, 1)
 

Normalized defining polynomial

\( x^{16} + 156 x^{14} + 11594 x^{12} - 24 x^{11} + 531680 x^{10} + 4032 x^{9} + 16317753 x^{8} + 247008 x^{7} + 340542452 x^{6} + 3145200 x^{5} + 4685677056 x^{4} - 43671120 x^{3} + 38637575756 x^{2} - 926997912 x + 145330457809 \)

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:  \(7269924201415415149428216006967296=2^{48}\cdot 3^{8}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.72$
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:  $2, 3, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4272=2^{4}\cdot 3\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{4272}(1,·)$, $\chi_{4272}(3205,·)$, $\chi_{4272}(1601,·)$, $\chi_{4272}(2315,·)$, $\chi_{4272}(1423,·)$, $\chi_{4272}(533,·)$, $\chi_{4272}(2137,·)$, $\chi_{4272}(3737,·)$, $\chi_{4272}(1247,·)$, $\chi_{4272}(355,·)$, $\chi_{4272}(3559,·)$, $\chi_{4272}(2669,·)$, $\chi_{4272}(1069,·)$, $\chi_{4272}(179,·)$, $\chi_{4272}(3383,·)$, $\chi_{4272}(2491,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{24565388057} a^{14} + \frac{11816024162}{24565388057} a^{13} - \frac{2725339222}{24565388057} a^{12} - \frac{1317399841}{3509341151} a^{11} + \frac{1474672309}{3509341151} a^{10} + \frac{1629233526}{24565388057} a^{9} + \frac{1429038500}{24565388057} a^{8} + \frac{672386829}{3509341151} a^{7} - \frac{6948249013}{24565388057} a^{6} + \frac{4803804401}{24565388057} a^{5} - \frac{11087644161}{24565388057} a^{4} - \frac{3933770251}{24565388057} a^{3} - \frac{7455819885}{24565388057} a^{2} + \frac{8981314045}{24565388057} a + \frac{7209684115}{24565388057}$, $\frac{1}{8482393351005386045295782617512319202572017209} a^{15} + \frac{151973181265872712566683807138010564}{8482393351005386045295782617512319202572017209} a^{14} + \frac{147254648743069214142404850922844841365301027}{8482393351005386045295782617512319202572017209} a^{13} - \frac{87242796155741228588358893074640041396612178}{180476454276710341389271970585368493671745047} a^{12} + \frac{56520357038825324219655792437106707565900390}{1211770478715055149327968945358902743224573887} a^{11} + \frac{715485407226392097109471038162277732924903288}{8482393351005386045295782617512319202572017209} a^{10} + \frac{2187901747351822978656270590974921289372400883}{8482393351005386045295782617512319202572017209} a^{9} - \frac{3299665860771100455550292808816238664810645337}{8482393351005386045295782617512319202572017209} a^{8} + \frac{1189283616216876312638337962897735807539476441}{8482393351005386045295782617512319202572017209} a^{7} + \frac{133628159470760836496388443570345883536831542}{8482393351005386045295782617512319202572017209} a^{6} - \frac{454401116035647972134813089458735845458188185}{8482393351005386045295782617512319202572017209} a^{5} + \frac{332182325231789090465288110166582902034255026}{8482393351005386045295782617512319202572017209} a^{4} - \frac{2233507490729093719869100105417801785681975844}{8482393351005386045295782617512319202572017209} a^{3} - \frac{3999397610190425462691690476033212414336104335}{8482393351005386045295782617512319202572017209} a^{2} - \frac{2689619551123901439836987312779279383230096140}{8482393351005386045295782617512319202572017209} a + \frac{3646312806230009347379965977782715983856355783}{8482393351005386045295782617512319202572017209}$

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

Class group and class number

$C_{4}\times C_{52}\times C_{6240}$, which has order $1297920$ (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:  \( 11964.310642723332 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-178}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-534}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-89}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-267}) \), \(\Q(\sqrt{3}, \sqrt{-178})\), \(\Q(\sqrt{2}, \sqrt{-89})\), \(\Q(\sqrt{6}, \sqrt{-178})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-89})\), \(\Q(\sqrt{2}, \sqrt{-267})\), \(\Q(\sqrt{6}, \sqrt{-89})\), 4.0.145999872.5, 4.4.18432.1, 4.0.16222208.5, \(\Q(\zeta_{16})^+\), 8.0.333061916000256.34, 8.0.85263850496065536.3, 8.0.1052640129581056.34, 8.0.85263850496065536.4, \(\Q(\zeta_{48})^+\), 8.0.21315962624016384.88, 8.0.21315962624016384.45

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$