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Properties

Label 16.0.23879602510...056.22
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 13^{12}\cdot 17^{12}$
Root discriminant $385.59$
Ramified primes $2, 13, 17$
Class number $119808000$ (GRH)
Class group $[2, 2, 2, 2, 4, 120, 15600]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4121239940624, -352153340032, 535089365184, -40748008704, 35413540288, -2216400576, 1387466400, -70097664, 36424504, -1465568, 638672, -20800, 8368, -208, 88, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 88*x^14 - 208*x^13 + 8368*x^12 - 20800*x^11 + 638672*x^10 - 1465568*x^9 + 36424504*x^8 - 70097664*x^7 + 1387466400*x^6 - 2216400576*x^5 + 35413540288*x^4 - 40748008704*x^3 + 535089365184*x^2 - 352153340032*x + 4121239940624)
 
gp: K = bnfinit(x^16 + 88*x^14 - 208*x^13 + 8368*x^12 - 20800*x^11 + 638672*x^10 - 1465568*x^9 + 36424504*x^8 - 70097664*x^7 + 1387466400*x^6 - 2216400576*x^5 + 35413540288*x^4 - 40748008704*x^3 + 535089365184*x^2 - 352153340032*x + 4121239940624, 1)
 

Normalized defining polynomial

\( x^{16} + 88 x^{14} - 208 x^{13} + 8368 x^{12} - 20800 x^{11} + 638672 x^{10} - 1465568 x^{9} + 36424504 x^{8} - 70097664 x^{7} + 1387466400 x^{6} - 2216400576 x^{5} + 35413540288 x^{4} - 40748008704 x^{3} + 535089365184 x^{2} - 352153340032 x + 4121239940624 \)

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:  \(238796025103060449638782189092971852333056=2^{44}\cdot 13^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $385.59$
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:  $2, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3536=2^{4}\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3536}(1,·)$, $\chi_{3536}(645,·)$, $\chi_{3536}(1665,·)$, $\chi_{3536}(2313,·)$, $\chi_{3536}(3229,·)$, $\chi_{3536}(1169,·)$, $\chi_{3536}(2517,·)$, $\chi_{3536}(441,·)$, $\chi_{3536}(2393,·)$, $\chi_{3536}(1565,·)$, $\chi_{3536}(837,·)$, $\chi_{3536}(421,·)$, $\chi_{3536}(1373,·)$, $\chi_{3536}(1585,·)$, $\chi_{3536}(2809,·)$, $\chi_{3536}(1789,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{64} a^{8} + \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a + \frac{7}{16}$, $\frac{1}{64} a^{9} + \frac{1}{16} a^{7} - \frac{1}{8} a^{6} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{7}{16} a$, $\frac{1}{64} a^{10} - \frac{1}{8} a^{7} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{7}{16} a^{2} + \frac{1}{4}$, $\frac{1}{2432} a^{11} - \frac{5}{1216} a^{9} + \frac{3}{608} a^{8} - \frac{11}{304} a^{7} + \frac{3}{76} a^{6} - \frac{25}{304} a^{5} - \frac{9}{152} a^{4} + \frac{131}{608} a^{3} - \frac{3}{38} a^{2} - \frac{121}{304} a + \frac{1}{8}$, $\frac{1}{96136960} a^{12} - \frac{537}{24034240} a^{11} - \frac{8213}{2403424} a^{10} + \frac{45691}{24034240} a^{9} + \frac{344401}{48068480} a^{8} + \frac{417489}{6008560} a^{7} - \frac{7975}{150214} a^{6} + \frac{7317}{375535} a^{5} - \frac{397933}{4806848} a^{4} - \frac{275977}{6008560} a^{3} + \frac{744429}{3004280} a^{2} - \frac{1058363}{6008560} a + \frac{143237}{632480}$, $\frac{1}{96136960} a^{13} - \frac{587}{48068480} a^{11} + \frac{131901}{24034240} a^{10} - \frac{3597}{2529920} a^{9} - \frac{153}{81472} a^{8} + \frac{329021}{3004280} a^{7} - \frac{138853}{6008560} a^{6} + \frac{41137}{358720} a^{5} - \frac{13949}{751070} a^{4} - \frac{4249941}{12017120} a^{3} - \frac{1758979}{6008560} a^{2} + \frac{436247}{2403424} a + \frac{124133}{316240}$, $\frac{1}{240634456740673762560} a^{14} + \frac{7250453557}{5013217848764036720} a^{13} - \frac{10821756265}{16042297116044917504} a^{12} + \frac{14296454301568909}{120317228370336881280} a^{11} - \frac{794794041145917361}{120317228370336881280} a^{10} - \frac{8266983902007357}{2005287139505614688} a^{9} + \frac{2678167689343529}{407855011424870784} a^{8} + \frac{167042649088039831}{1503965354629211016} a^{7} - \frac{700812006864386991}{20052871395056146880} a^{6} + \frac{756082052492510267}{15039653546292110160} a^{5} + \frac{412896354196456297}{4010574279011229376} a^{4} + \frac{2750963110401091639}{6015861418516844064} a^{3} + \frac{9206484678896748701}{30079307092584220320} a^{2} - \frac{82245266344994021}{187995669328651377} a - \frac{668962806848242901}{1583121425925485280}$, $\frac{1}{2530983223798414958912937996111360} a^{15} + \frac{362189091349}{210915268649867913242744833009280} a^{14} + \frac{5344557940145174207027}{1647775536327093072208944007885} a^{13} - \frac{61183840990493610178133}{16651205419726414203374592079680} a^{12} + \frac{32298677552909889100796499413}{1265491611899207479456468998055680} a^{11} + \frac{192425824207314153071081098581}{26364408581233489155343104126160} a^{10} - \frac{2338334979995365719059832544297}{316372902974801869864117249513920} a^{9} + \frac{348931239499910786886481136261}{79093225743700467466029312378480} a^{8} + \frac{9729902266870089057283600063}{3147989084326386764817087059840} a^{7} + \frac{7633938638596006827436640888453}{158186451487400934932058624756960} a^{6} + \frac{600518970986443750608979224629}{6591102145308372288835776031540} a^{5} + \frac{1210464241128605061682980111643}{15818645148740093493205862475696} a^{4} + \frac{50835639455694401962436327224067}{316372902974801869864117249513920} a^{3} - \frac{328029889657945475936845433738}{4943326608981279216626832023655} a^{2} - \frac{6558267100608082429616016905419}{15818645148740093493205862475696} a - \frac{1924053177469701468422015287}{69380022582193392514060800332}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{120}\times C_{15600}$, which has order $119808000$ (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:  \( 27023332.036477774 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{442}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{17}) \), 4.0.22105827328.8, \(\Q(\sqrt{17}, \sqrt{26})\), 4.0.22105827328.7, 4.0.4499456.2, 4.0.1300342784.10, 4.4.314432.1, 4.4.830297.1, 8.0.488667601855351619584.1, 8.0.1690891355900870656.3, 8.8.2823754171224064.9

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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$2.4.11.8$x^{4} + 4 x^{2} + 10$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.8$x^{4} + 4 x^{2} + 10$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.8$x^{4} + 4 x^{2} + 10$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.8$x^{4} + 4 x^{2} + 10$$4$$1$$11$$C_4$$[3, 4]$
$13$13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$