Properties

Label 16.8.255...841.1
Degree $16$
Signature $[8, 4]$
Discriminant $2.552\times 10^{46}$
Root discriminant $795.11$
Ramified primes $41, 71$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 13*x^14 + 2974*x^13 - 264473*x^12 + 795750*x^11 - 49905603*x^10 - 35476954*x^9 - 111581072*x^8 + 12241189350*x^7 + 316573419487*x^6 + 371149914965*x^5 + 332497946616*x^4 - 33962432992424*x^3 - 230531059041744*x^2 + 709344561065266*x - 463451459871559)
 
gp: K = bnfinit(x^16 - 4*x^15 - 13*x^14 + 2974*x^13 - 264473*x^12 + 795750*x^11 - 49905603*x^10 - 35476954*x^9 - 111581072*x^8 + 12241189350*x^7 + 316573419487*x^6 + 371149914965*x^5 + 332497946616*x^4 - 33962432992424*x^3 - 230531059041744*x^2 + 709344561065266*x - 463451459871559, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-463451459871559, 709344561065266, -230531059041744, -33962432992424, 332497946616, 371149914965, 316573419487, 12241189350, -111581072, -35476954, -49905603, 795750, -264473, 2974, -13, -4, 1]);
 

\( x^{16} - 4 x^{15} - 13 x^{14} + 2974 x^{13} - 264473 x^{12} + 795750 x^{11} - 49905603 x^{10} - 35476954 x^{9} - 111581072 x^{8} + 12241189350 x^{7} + 316573419487 x^{6} + 371149914965 x^{5} + 332497946616 x^{4} - 33962432992424 x^{3} - 230531059041744 x^{2} + 709344561065266 x - 463451459871559 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(25518669979516247217450329195440215991007686841\)\(\medspace = 41^{15}\cdot 71^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $795.11$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $41, 71$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{23} a^{13} + \frac{2}{23} a^{12} + \frac{7}{23} a^{11} - \frac{11}{23} a^{10} + \frac{3}{23} a^{9} - \frac{8}{23} a^{8} + \frac{11}{23} a^{7} - \frac{2}{23} a^{6} - \frac{5}{23} a^{5} - \frac{10}{23} a^{4} + \frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{2}{23} a + \frac{5}{23}$, $\frac{1}{116219} a^{14} + \frac{770}{116219} a^{13} - \frac{54278}{116219} a^{12} + \frac{55988}{116219} a^{11} + \frac{4297}{116219} a^{10} + \frac{41833}{116219} a^{9} - \frac{54525}{116219} a^{8} + \frac{40485}{116219} a^{7} + \frac{337}{5053} a^{6} - \frac{29127}{116219} a^{5} - \frac{44800}{116219} a^{4} - \frac{29764}{116219} a^{3} + \frac{11897}{116219} a^{2} + \frac{1242}{5053} a - \frac{55431}{116219}$, $\frac{1}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a^{15} - \frac{13744251236135093488392389310298501329013665557066559559690894655471768170954148341483262692885713596}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a^{14} + \frac{60905867831003163323729643678019437890517118804752959744957686815083404465461507798382560105318898636595}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a^{13} - \frac{1001059181937947681207346014652608633951842006607340183093222886153446059808728812754595034371030032639517}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a^{12} + \frac{28385092305619040006205671795492569685280106473011140891623096761033959127644927140044371917750162119176}{95049770915691666067813014352058725297346737367605610507602578563406606427977174493869399016734688963067} a^{11} + \frac{1671926461050571382752340515882277941434243668286085983663675866970057820636196668007089819347407936024658}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a^{10} - \frac{1617726115394388801212349420282027081719721615021519949298147929803581929925917882035513219226799235377528}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a^{9} + \frac{1165928532006474979439968214590172937091672567843206540646681802816823725119094325362198962818136245621887}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a^{8} + \frac{910944635777762073429094429780822802232298885234225712216028346693296970090835340452549852920593174462230}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a^{7} + \frac{1636710571930898025524057214246121023159426608528209761937608827541627312937608820188870021174292142323413}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a^{6} + \frac{99168407656520696575278043015062465087367668313936945071543843184994667603744136372062823401501369571500}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a^{5} + \frac{24403870420048822781811797532029925934046863326358379539932838443499814899558875021352806871669506303429}{152906153212199636717786153522877079826166490547887286468751974210697584253702411142311641896486238766673} a^{4} + \frac{1915998615607074787931345584908568129439176110709970509971799515029936362655516702797747820056004354597}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a^{3} + \frac{1534670155422457160907991461406423927986945809848897327955752814479836427238543340261053858865005442195402}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a^{2} + \frac{172505029294874389876564366377013949459366191643926763897581075930657260606146651415636457093536600581343}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479} a + \frac{907405955573525066138223617549609451121981013015812206664634751743295876640067309008868067219990209199478}{3516841523880591644509081531026172836001829282601407588781295406846044437835155456273167763619183491633479}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 222832973520000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 222832973520000000 \cdot 1}{2\sqrt{25518669979516247217450329195440215991007686841}}\approx 0.278278645619334$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.4949033481250603961.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
41Data not computed
71Data not computed