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Properties

Label 16.0.21767174858...0161.3
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 157^{14}$
Root discriminant $787.25$
Ramified primes $13, 157$
Class number $6051600$ (GRH)
Class group $[2, 1230, 2460]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![82946223837577, -38055428105350, -3454592429488, 5076352091595, 456414116225, -249046344, 15572551375, -6294591692, 1476282219, -163195065, 11768472, -340621, 28306, -627, 187, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 187*x^14 - 627*x^13 + 28306*x^12 - 340621*x^11 + 11768472*x^10 - 163195065*x^9 + 1476282219*x^8 - 6294591692*x^7 + 15572551375*x^6 - 249046344*x^5 + 456414116225*x^4 + 5076352091595*x^3 - 3454592429488*x^2 - 38055428105350*x + 82946223837577)
 
gp: K = bnfinit(x^16 - 3*x^15 + 187*x^14 - 627*x^13 + 28306*x^12 - 340621*x^11 + 11768472*x^10 - 163195065*x^9 + 1476282219*x^8 - 6294591692*x^7 + 15572551375*x^6 - 249046344*x^5 + 456414116225*x^4 + 5076352091595*x^3 - 3454592429488*x^2 - 38055428105350*x + 82946223837577, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 187 x^{14} - 627 x^{13} + 28306 x^{12} - 340621 x^{11} + 11768472 x^{10} - 163195065 x^{9} + 1476282219 x^{8} - 6294591692 x^{7} + 15572551375 x^{6} - 249046344 x^{5} + 456414116225 x^{4} + 5076352091595 x^{3} - 3454592429488 x^{2} - 38055428105350 x + 82946223837577 \)

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:  \(21767174858780577201904716993499183222663790161=13^{14}\cdot 157^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $787.25$
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:  $13, 157$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2770} a^{14} - \frac{13}{277} a^{13} - \frac{281}{1385} a^{12} + \frac{41}{1385} a^{11} + \frac{29}{277} a^{10} - \frac{499}{2770} a^{9} + \frac{7}{277} a^{8} - \frac{392}{1385} a^{7} + \frac{231}{1385} a^{6} - \frac{27}{277} a^{5} - \frac{1373}{2770} a^{4} + \frac{426}{1385} a^{3} - \frac{361}{1385} a^{2} - \frac{462}{1385} a - \frac{671}{1385}$, $\frac{1}{66701702118548196764499085983937253328952402837602557525019696178359518521903791150111404275532422370} a^{15} + \frac{3172026508405117356765006414356707426908120497215700273487473568029619292565171269222983876779353}{33350851059274098382249542991968626664476201418801278762509848089179759260951895575055702137766211185} a^{14} + \frac{1937351995890562731902680532276312076309916776154306890815370352293890539582945414532467158098447649}{33350851059274098382249542991968626664476201418801278762509848089179759260951895575055702137766211185} a^{13} - \frac{2410434932778578890534348943611414049250082074982437564803116791433491408179022153807425993300175235}{13340340423709639352899817196787450665790480567520511505003939235671903704380758230022280855106484474} a^{12} - \frac{12165797028904501120428676645135404992463015863723543857023174967491975154622341781357788381476918503}{66701702118548196764499085983937253328952402837602557525019696178359518521903791150111404275532422370} a^{11} - \frac{3910982618678182668308604623282958762638491236364708363876180362735689369179674883349709593601979519}{66701702118548196764499085983937253328952402837602557525019696178359518521903791150111404275532422370} a^{10} + \frac{16593331758249311979168090748793690697445828196730735590942094687888835512753218720552233576654328993}{33350851059274098382249542991968626664476201418801278762509848089179759260951895575055702137766211185} a^{9} + \frac{3525652427935447881978995927067470973764205400798438216244117938056907638773030023960378449918027893}{33350851059274098382249542991968626664476201418801278762509848089179759260951895575055702137766211185} a^{8} - \frac{32457393326086028507666257744247788432339978034503194441441737349993796477276433466491856867561416457}{66701702118548196764499085983937253328952402837602557525019696178359518521903791150111404275532422370} a^{7} - \frac{3793929523223838458211603793207282603455507495566461982433205845257151567546072369797990618282692363}{66701702118548196764499085983937253328952402837602557525019696178359518521903791150111404275532422370} a^{6} + \frac{29314469309613126275316913188036107764214702796959086751466871860240838969446152352629272734757900727}{66701702118548196764499085983937253328952402837602557525019696178359518521903791150111404275532422370} a^{5} - \frac{12397285709786426251252428884691064354963975150103292608905816620393113062972380619680452664737214668}{33350851059274098382249542991968626664476201418801278762509848089179759260951895575055702137766211185} a^{4} - \frac{2099205323968331257392406192036254588789622173414967310762485867313250090373036095863423834922466006}{6670170211854819676449908598393725332895240283760255752501969617835951852190379115011140427553242237} a^{3} + \frac{29931567171118769574067163877413767348818974090916347562287099599148526363471113289539395060022064839}{66701702118548196764499085983937253328952402837602557525019696178359518521903791150111404275532422370} a^{2} - \frac{19197267792848630007454779320820603707832197593731906457909329420703347464843440913585841034310839071}{66701702118548196764499085983937253328952402837602557525019696178359518521903791150111404275532422370} a - \frac{5992404190654330272847831907742770615606021470791983996506129354514348134376217677681193587502352541}{33350851059274098382249542991968626664476201418801278762509848089179759260951895575055702137766211185}$

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

Class group and class number

$C_{2}\times C_{1230}\times C_{2460}$, which has order $6051600$ (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:  \( 3701634329.13 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{2041}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{157}) \), 4.4.8502154921.2, 4.4.8502154921.1, \(\Q(\sqrt{13}, \sqrt{157})\), 8.8.72286638300684516241.3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$157$157.8.7.2$x^{8} - 3925$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
157.8.7.2$x^{8} - 3925$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$