↑

Properties

Label 16.8.90087898081...1561.3
Degree $16$
Signature $[8, 4]$
Discriminant $53^{14}\cdot 97^{14}$
Root discriminant $1766.72$
Ramified primes $53, 97$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26882036630663087, -11643515053935122, -30969923082952135, -12848087213470847, -1430607533707472, 18731643146726, 3340517012461, 345679229513, 43904988053, -139487100, -13645887, -1484709, -404736, 161, -41, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 41*x^14 + 161*x^13 - 404736*x^12 - 1484709*x^11 - 13645887*x^10 - 139487100*x^9 + 43904988053*x^8 + 345679229513*x^7 + 3340517012461*x^6 + 18731643146726*x^5 - 1430607533707472*x^4 - 12848087213470847*x^3 - 30969923082952135*x^2 - 11643515053935122*x + 26882036630663087)
 
gp: K = bnfinit(x^16 - 4*x^15 - 41*x^14 + 161*x^13 - 404736*x^12 - 1484709*x^11 - 13645887*x^10 - 139487100*x^9 + 43904988053*x^8 + 345679229513*x^7 + 3340517012461*x^6 + 18731643146726*x^5 - 1430607533707472*x^4 - 12848087213470847*x^3 - 30969923082952135*x^2 - 11643515053935122*x + 26882036630663087, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 41 x^{14} + 161 x^{13} - 404736 x^{12} - 1484709 x^{11} - 13645887 x^{10} - 139487100 x^{9} + 43904988053 x^{8} + 345679229513 x^{7} + 3340517012461 x^{6} + 18731643146726 x^{5} - 1430607533707472 x^{4} - 12848087213470847 x^{3} - 30969923082952135 x^{2} - 11643515053935122 x + 26882036630663087 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9008789808170658548806656862365268212872206896031561=53^{14}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1766.72$
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:  $53, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{97} a^{8} - \frac{2}{97} a^{7} + \frac{26}{97} a^{6} - \frac{13}{97} a^{5} - \frac{2}{97} a^{4} - \frac{19}{97} a^{3} - \frac{34}{97} a^{2} + \frac{44}{97} a + \frac{35}{97}$, $\frac{1}{291} a^{9} + \frac{1}{291} a^{8} + \frac{39}{97} a^{7} - \frac{43}{97} a^{6} + \frac{56}{291} a^{5} - \frac{25}{291} a^{4} + \frac{2}{97} a^{3} + \frac{13}{97} a^{2} - \frac{124}{291} a + \frac{8}{291}$, $\frac{1}{873} a^{10} - \frac{1}{873} a^{9} + \frac{4}{873} a^{8} + \frac{50}{291} a^{7} + \frac{47}{873} a^{6} + \frac{433}{873} a^{5} + \frac{278}{873} a^{4} + \frac{130}{291} a^{3} - \frac{211}{873} a^{2} + \frac{319}{873} a + \frac{173}{873}$, $\frac{1}{2619} a^{11} + \frac{1}{873} a^{9} + \frac{1}{2619} a^{8} - \frac{1243}{2619} a^{7} + \frac{289}{873} a^{6} - \frac{88}{291} a^{5} + \frac{974}{2619} a^{4} - \frac{406}{2619} a^{3} - \frac{89}{291} a^{2} - \frac{334}{873} a + \frac{56}{2619}$, $\frac{1}{738558} a^{12} + \frac{17}{369279} a^{11} + \frac{91}{246186} a^{10} - \frac{70}{369279} a^{9} + \frac{497}{246186} a^{8} + \frac{156679}{369279} a^{7} + \frac{18020}{123093} a^{6} + \frac{218423}{738558} a^{5} - \frac{27925}{369279} a^{4} + \frac{298919}{738558} a^{3} - \frac{30473}{123093} a^{2} - \frac{224227}{738558} a - \frac{7805}{15714}$, $\frac{1}{6245985006} a^{13} - \frac{1052}{3122992503} a^{12} - \frac{515723}{6245985006} a^{11} - \frac{1710649}{3122992503} a^{10} + \frac{6239731}{6245985006} a^{9} + \frac{1140562}{3122992503} a^{8} - \frac{747447509}{3122992503} a^{7} + \frac{956430707}{6245985006} a^{6} + \frac{57253652}{1040997501} a^{5} + \frac{42467575}{693998334} a^{4} + \frac{551981473}{3122992503} a^{3} + \frac{463935173}{6245985006} a^{2} - \frac{456288497}{6245985006} a + \frac{14267174}{66446649}$, $\frac{1}{268818618920826762} a^{14} + \frac{44485}{953257513903641} a^{13} + \frac{32610241337}{89606206306942254} a^{12} - \frac{5423019886790}{134409309460413381} a^{11} - \frac{51444418259977}{268818618920826762} a^{10} + \frac{3629021525584}{44803103153471127} a^{9} - \frac{159899724458596}{134409309460413381} a^{8} - \frac{45347434290457397}{268818618920826762} a^{7} + \frac{34010631059278747}{134409309460413381} a^{6} + \frac{24171756647876017}{89606206306942254} a^{5} - \frac{38868111442459955}{134409309460413381} a^{4} - \frac{37059837637352525}{268818618920826762} a^{3} + \frac{12889715948860583}{89606206306942254} a^{2} - \frac{11865351905525125}{44803103153471127} a + \frac{1258215103832090}{2859772541710923}$, $\frac{1}{873292543158235361115682366400591805474368352042156690412773055101404793284444814} a^{15} - \frac{103159755861970612886416353473091638186947653885613276290317735}{873292543158235361115682366400591805474368352042156690412773055101404793284444814} a^{14} + \frac{1263266665354555861569470210160822654110251514035324767206418872247341}{16172084132559914094734858637047996397673488000780679452088389909285273949711941} a^{13} - \frac{17883184504789085561579675029804169308381679227371779366217731330186651930}{436646271579117680557841183200295902737184176021078345206386527550702396642222407} a^{12} - \frac{24077691681796712747271081945056731204465263449908222510130991767252157224470}{436646271579117680557841183200295902737184176021078345206386527550702396642222407} a^{11} + \frac{220757366163066821870314540365369882325513413637176471096349361804239365134629}{436646271579117680557841183200295902737184176021078345206386527550702396642222407} a^{10} - \frac{891994169386745215707966298367529841130643556528637758323514335935639657439545}{873292543158235361115682366400591805474368352042156690412773055101404793284444814} a^{9} + \frac{1351936059398158645307613739413221276197168825783112082589706541399945367151317}{436646271579117680557841183200295902737184176021078345206386527550702396642222407} a^{8} - \frac{27568907393636617569766328740710519182107884280054217991782277463414054652195021}{97032504795359484568409151822287978386040928004684076712530339455711643698271646} a^{7} - \frac{52952548217004061333081736443494613024551937338879692913681369881716585987376936}{436646271579117680557841183200295902737184176021078345206386527550702396642222407} a^{6} + \frac{197996111849109164289184351602258579395147584726404953552665085898213108581302159}{436646271579117680557841183200295902737184176021078345206386527550702396642222407} a^{5} + \frac{26572113261331146697591570004324014847728673840006083923221659714101617182318582}{145548757193039226852613727733431967579061392007026115068795509183567465547407469} a^{4} + \frac{123388979586040658502137097442994648317495141662793886833017214535857655434111351}{873292543158235361115682366400591805474368352042156690412773055101404793284444814} a^{3} + \frac{665063179582082226803506098963823076801398883799847529716276260620948290967620}{48516252397679742284204575911143989193020464002342038356265169727855821849135823} a^{2} - \frac{198606415081992728134412289779922066976811345116647269958720119156775474680596994}{436646271579117680557841183200295902737184176021078345206386527550702396642222407} a - \frac{5685973575530906979490310535250373233105736698729668994426699582904413420237997}{18580692407622028959908135455331740542007837277492695540697299044710740282647762}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $11$
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:  \( 22257618600600000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{5141}) \), \(\Q(\sqrt{53}, \sqrt{97})\), 8.8.18462292327593524004841.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$53$53.8.7.1$x^{8} - 53$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.1$x^{8} - 53$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$97$97.8.7.4$x^{8} - 1515625$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.4$x^{8} - 1515625$$8$$1$$7$$C_8$$[\ ]_{8}$