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Properties

Label 18.0.13052029345...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{18}\cdot 5^{9}\cdot 7^{15}\cdot 11^{9}\cdot 487^{9}$
Root discriminant $2484.94$
Ramified primes $3, 5, 7, 11, 487$
Class number $27606915072$ (GRH)
Class group $[2, 2, 2, 6, 6, 6, 246, 64944]$ (GRH)
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![421748783514829120, -6266059191729120, 17233229874363636, 8578727481828, 575675484948417, 8756629556136, 12034816391578, 139597980018, 200774929287, 729775284, 2469822387, -428442, 22856695, -8484, 153015, -42, 606, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 606*x^16 - 42*x^15 + 153015*x^14 - 8484*x^13 + 22856695*x^12 - 428442*x^11 + 2469822387*x^10 + 729775284*x^9 + 200774929287*x^8 + 139597980018*x^7 + 12034816391578*x^6 + 8756629556136*x^5 + 575675484948417*x^4 + 8578727481828*x^3 + 17233229874363636*x^2 - 6266059191729120*x + 421748783514829120)
 
gp: K = bnfinit(x^18 + 606*x^16 - 42*x^15 + 153015*x^14 - 8484*x^13 + 22856695*x^12 - 428442*x^11 + 2469822387*x^10 + 729775284*x^9 + 200774929287*x^8 + 139597980018*x^7 + 12034816391578*x^6 + 8756629556136*x^5 + 575675484948417*x^4 + 8578727481828*x^3 + 17233229874363636*x^2 - 6266059191729120*x + 421748783514829120, 1)
 

Normalized defining polynomial

\( x^{18} + 606 x^{16} - 42 x^{15} + 153015 x^{14} - 8484 x^{13} + 22856695 x^{12} - 428442 x^{11} + 2469822387 x^{10} + 729775284 x^{9} + 200774929287 x^{8} + 139597980018 x^{7} + 12034816391578 x^{6} + 8756629556136 x^{5} + 575675484948417 x^{4} + 8578727481828 x^{3} + 17233229874363636 x^{2} - 6266059191729120 x + 421748783514829120 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-13052029345071760863843934415776181167639406996071732693359375=-\,3^{18}\cdot 5^{9}\cdot 7^{15}\cdot 11^{9}\cdot 487^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2484.94$
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:  $3, 5, 7, 11, 487$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{28} a^{9} + \frac{1}{14} a^{7} + \frac{3}{14} a^{5} - \frac{1}{28} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{112} a^{10} + \frac{1}{112} a^{9} - \frac{5}{112} a^{8} - \frac{3}{28} a^{7} + \frac{3}{56} a^{6} + \frac{3}{56} a^{5} - \frac{15}{112} a^{4} - \frac{15}{112} a^{3} - \frac{7}{16} a^{2} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{112} a^{11} - \frac{1}{56} a^{9} - \frac{1}{16} a^{8} - \frac{1}{56} a^{7} - \frac{1}{4} a^{6} - \frac{25}{112} a^{5} - \frac{1}{4} a^{4} - \frac{5}{56} a^{3} - \frac{7}{16} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{56224} a^{12} + \frac{3}{56224} a^{10} - \frac{265}{28112} a^{9} + \frac{5405}{56224} a^{8} + \frac{3}{28} a^{7} - \frac{1451}{56224} a^{6} + \frac{5809}{28112} a^{5} - \frac{6805}{56224} a^{4} - \frac{5415}{14056} a^{3} + \frac{1271}{8032} a^{2} + \frac{467}{4016} a + \frac{485}{1004}$, $\frac{1}{112448} a^{13} - \frac{499}{112448} a^{11} - \frac{1}{4016} a^{10} - \frac{1121}{112448} a^{9} + \frac{1}{16} a^{8} - \frac{8479}{112448} a^{7} - \frac{963}{8032} a^{6} + \frac{2733}{112448} a^{5} + \frac{1931}{8032} a^{4} - \frac{55861}{112448} a^{3} - \frac{2545}{8032} a^{2} + \frac{92}{251} a - \frac{1}{2}$, $\frac{1}{224896} a^{14} + \frac{1}{224896} a^{12} - \frac{1}{8032} a^{11} - \frac{625}{224896} a^{10} + \frac{129}{14056} a^{9} + \frac{289}{224896} a^{8} - \frac{4733}{112448} a^{7} + \frac{303}{32128} a^{6} + \frac{15453}{112448} a^{5} + \frac{42587}{224896} a^{4} + \frac{38483}{112448} a^{3} + \frac{183}{1004} a^{2} + \frac{197}{2008} a + \frac{67}{502}$, $\frac{1}{224896} a^{15} - \frac{1}{224896} a^{13} + \frac{373}{224896} a^{11} + \frac{7}{8032} a^{10} + \frac{1747}{224896} a^{9} + \frac{1815}{16064} a^{8} + \frac{19079}{224896} a^{7} + \frac{371}{16064} a^{6} - \frac{433}{32128} a^{5} - \frac{1791}{16064} a^{4} + \frac{5041}{112448} a^{3} + \frac{231}{502} a^{2} - \frac{1181}{4016} a + \frac{383}{1004}$, $\frac{1}{63920306930841387487973795559913086888704} a^{16} - \frac{121124982053396372368974243041158181}{63920306930841387487973795559913086888704} a^{15} + \frac{6985350216488707970436755784412053}{4565736209345813391998128254279506206336} a^{14} + \frac{112556837345986569229751541629616761}{63920306930841387487973795559913086888704} a^{13} + \frac{16818777070528727802795086173971531}{1997509591588793358999181111247283965272} a^{12} - \frac{19387837198225479237565560783193986381}{63920306930841387487973795559913086888704} a^{11} + \frac{3606185370229830842717316034701344717}{15980076732710346871993448889978271722176} a^{10} + \frac{216751932247780555533351053177249991031}{63920306930841387487973795559913086888704} a^{9} - \frac{486950466059173914526145082883580880681}{3995019183177586717998362222494567930544} a^{8} - \frac{5652565792085122604537073948979583213131}{63920306930841387487973795559913086888704} a^{7} + \frac{5522097680385290279895842871595049865841}{31960153465420693743986897779956543444352} a^{6} - \frac{11180905881595952319628124689658530014965}{63920306930841387487973795559913086888704} a^{5} + \frac{1740081251964970332724624494003213174877}{63920306930841387487973795559913086888704} a^{4} - \frac{7699484675085916628838773896912498172741}{15980076732710346871993448889978271722176} a^{3} - \frac{225122656911796243142210446683477985539}{2282868104672906695999064127139753103168} a^{2} - \frac{25574901319630955258804707153092939835}{142679256542056668499941507946234568948} a - \frac{63519125149334580975290921536945588709}{142679256542056668499941507946234568948}$, $\frac{1}{144268629307753156606979528820790906461554320353246091665112832} a^{17} + \frac{918434944481649539333}{144268629307753156606979528820790906461554320353246091665112832} a^{16} - \frac{74667625008051110207790629538674432972549559738938650763}{36067157326938289151744882205197726615388580088311522916278208} a^{15} + \frac{40893772127519825360392633904261066436660971843272933931}{20609804186821879515282789831541558065936331479035155952158976} a^{14} - \frac{264388220646526169683077741287542944058654985947389593567}{72134314653876578303489764410395453230777160176623045832556416} a^{13} - \frac{41408315476210768958652688442854327055068129928312549091}{20609804186821879515282789831541558065936331479035155952158976} a^{12} - \frac{73859341629865645077226891435911102172464875582653877887315}{72134314653876578303489764410395453230777160176623045832556416} a^{11} - \frac{269188325039387010138093989656254339092766260540466831868665}{144268629307753156606979528820790906461554320353246091665112832} a^{10} + \frac{772222009361567780264240854231084872683816427125659789211415}{72134314653876578303489764410395453230777160176623045832556416} a^{9} - \frac{11936347258576300503406599718909791425449769572224181262811187}{144268629307753156606979528820790906461554320353246091665112832} a^{8} + \frac{228565590421747962360310427397604373561043269718274699200347}{36067157326938289151744882205197726615388580088311522916278208} a^{7} - \frac{627387247251953161853227593694249994687068141505003329504025}{144268629307753156606979528820790906461554320353246091665112832} a^{6} - \frac{29382721994075891469623696025279044321928975447631426766392061}{144268629307753156606979528820790906461554320353246091665112832} a^{5} - \frac{15619139696202128007795386995283093122358627227895557883956029}{72134314653876578303489764410395453230777160176623045832556416} a^{4} + \frac{759761775822931280568726783106178280817448256035172662819541}{5152451046705469878820697457885389516484082869758788988039744} a^{3} - \frac{584901921931500020292857200998246446126117384413770220571437}{2576225523352734939410348728942694758242041434879394494019872} a^{2} + \frac{21151435305955127922776857487258877222147680134788882007209}{161014095209545933713146795558918422390127589679962155876242} a + \frac{10106236259299180339442924770462929074601770854143490335741}{161014095209545933713146795558918422390127589679962155876242}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{6}\times C_{246}\times C_{64944}$, which has order $27606915072$ (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:  $8$
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:  \( 5044715737238670.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-187495}) \), 3.1.187495.1 x3, 3.1.1323.1, Deg 6, 6.0.6591269545312375.1, 9.1.311496036263380728577125.1 x3

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{6}$ R R R R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
487Data not computed