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Properties

Label 15.1.18687424919...7504.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{14}\cdot 11^{7}\cdot 911^{4}\cdot 1181^{4}\cdot 2570851^{4}$
Root discriminant $12{,}155.42$
Ramified primes $2, 11, 911, 1181, 2570851$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group 15T48

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2414037401865775754839444, -450826849981549178240020, -47733821804558780746772, -3394681526819160320279, -173115997374755324728, -6359731734673496810, -162578031975372552, -2566914340949249, -11160258257892, 533228094032, 13103162268, 86964215, -1133008, -18994, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 18994*x^13 - 1133008*x^12 + 86964215*x^11 + 13103162268*x^10 + 533228094032*x^9 - 11160258257892*x^8 - 2566914340949249*x^7 - 162578031975372552*x^6 - 6359731734673496810*x^5 - 173115997374755324728*x^4 - 3394681526819160320279*x^3 - 47733821804558780746772*x^2 - 450826849981549178240020*x - 2414037401865775754839444)
 
gp: K = bnfinit(x^15 - 18994*x^13 - 1133008*x^12 + 86964215*x^11 + 13103162268*x^10 + 533228094032*x^9 - 11160258257892*x^8 - 2566914340949249*x^7 - 162578031975372552*x^6 - 6359731734673496810*x^5 - 173115997374755324728*x^4 - 3394681526819160320279*x^3 - 47733821804558780746772*x^2 - 450826849981549178240020*x - 2414037401865775754839444, 1)
 

Normalized defining polynomial

\( x^{15} - 18994 x^{13} - 1133008 x^{12} + 86964215 x^{11} + 13103162268 x^{10} + 533228094032 x^{9} - 11160258257892 x^{8} - 2566914340949249 x^{7} - 162578031975372552 x^{6} - 6359731734673496810 x^{5} - 173115997374755324728 x^{4} - 3394681526819160320279 x^{3} - 47733821804558780746772 x^{2} - 450826849981549178240020 x - 2414037401865775754839444 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18687424919832592866038564565999538838928123748131355638677504=-\,2^{14}\cdot 11^{7}\cdot 911^{4}\cdot 1181^{4}\cdot 2570851^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12{,}155.42$
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:  $2, 11, 911, 1181, 2570851$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{176} a^{8} - \frac{9}{176} a^{7} - \frac{5}{88} a^{6} + \frac{7}{88} a^{5} - \frac{19}{176} a^{4} + \frac{59}{176} a^{3} - \frac{1}{22} a^{2} + \frac{2}{11} a + \frac{13}{44}$, $\frac{1}{704} a^{9} - \frac{1}{352} a^{8} - \frac{29}{704} a^{7} - \frac{3}{176} a^{6} - \frac{141}{704} a^{5} - \frac{15}{352} a^{4} - \frac{299}{704} a^{3} - \frac{7}{44} a^{2} + \frac{69}{176} a + \frac{3}{176}$, $\frac{1}{704} a^{10} - \frac{1}{704} a^{8} - \frac{3}{352} a^{7} + \frac{43}{704} a^{6} - \frac{5}{88} a^{5} - \frac{87}{704} a^{4} + \frac{149}{352} a^{3} - \frac{51}{176} a^{2} + \frac{45}{176} a + \frac{35}{88}$, $\frac{1}{1408} a^{11} - \frac{29}{704} a^{7} + \frac{1}{32} a^{6} + \frac{15}{352} a^{5} - \frac{15}{352} a^{4} + \frac{497}{1408} a^{3} + \frac{1}{352} a^{2} - \frac{149}{352} a + \frac{107}{352}$, $\frac{1}{5632} a^{12} + \frac{1}{5632} a^{11} - \frac{1}{1408} a^{9} + \frac{7}{2816} a^{8} + \frac{115}{2816} a^{7} + \frac{71}{704} a^{6} + \frac{277}{1408} a^{5} + \frac{397}{5632} a^{4} + \frac{897}{5632} a^{3} + \frac{91}{352} a^{2} - \frac{255}{704} a + \frac{575}{1408}$, $\frac{1}{11264} a^{13} - \frac{1}{11264} a^{11} - \frac{1}{2816} a^{10} + \frac{1}{5632} a^{9} - \frac{1}{1408} a^{8} + \frac{145}{5632} a^{7} - \frac{233}{2816} a^{6} + \frac{2185}{11264} a^{5} - \frac{299}{2816} a^{4} + \frac{95}{11264} a^{3} + \frac{43}{1408} a^{2} - \frac{659}{2816} a - \frac{1135}{2816}$, $\frac{1}{66885843564103432215688508078566334610119874037711946064044975452251542716435140758282334009520258001356717215875351138187144697856} a^{14} + \frac{693912092746242928023543958067658339587556064821697616763131805901168669117428472904091511072183893258298480712014920818847471}{66885843564103432215688508078566334610119874037711946064044975452251542716435140758282334009520258001356717215875351138187144697856} a^{13} + \frac{1236397031857574250211888349106141512376377872051149247130089719633345404427611489329198680135898663671701414448150772880550959}{66885843564103432215688508078566334610119874037711946064044975452251542716435140758282334009520258001356717215875351138187144697856} a^{12} - \frac{20958365391272880655483359707929642947406854529346749379062492227764575413256460036074443250482897701433571070475098235315936499}{66885843564103432215688508078566334610119874037711946064044975452251542716435140758282334009520258001356717215875351138187144697856} a^{11} + \frac{2277929989736184711453671100698283336257555221267412495070896214904131969762199789240228152872594764201934993908061173299365699}{33442921782051716107844254039283167305059937018855973032022487726125771358217570379141167004760129000678358607937675569093572348928} a^{10} - \frac{10242802371838306133446542705151207464831723005851369786718633748742183092455870174825530761196309369725680593279231944223224981}{33442921782051716107844254039283167305059937018855973032022487726125771358217570379141167004760129000678358607937675569093572348928} a^{9} + \frac{93333017387890749384449324871463360706451013654289945129559484097625793101769128327532078743643989071363565555126488657253541253}{33442921782051716107844254039283167305059937018855973032022487726125771358217570379141167004760129000678358607937675569093572348928} a^{8} - \frac{870093373380778762320566353279642746168579358722722604579819980474267640283797778547920014019358364682274814549229153963139381331}{33442921782051716107844254039283167305059937018855973032022487726125771358217570379141167004760129000678358607937675569093572348928} a^{7} - \frac{6997171723905914371091458055013934356546848623350968660252926705016687255491379805857619140795210381883980030666904183192827212691}{66885843564103432215688508078566334610119874037711946064044975452251542716435140758282334009520258001356717215875351138187144697856} a^{6} - \frac{4567205253679705181681072750852410715371185761156216590085352260136794476560084699912401742398452830001434393898937388159191497989}{66885843564103432215688508078566334610119874037711946064044975452251542716435140758282334009520258001356717215875351138187144697856} a^{5} + \frac{16518988256292019290848787471071997702815075505163043376708286987211075836157882574122789183851354280201632757769147766999680952507}{66885843564103432215688508078566334610119874037711946064044975452251542716435140758282334009520258001356717215875351138187144697856} a^{4} + \frac{18071940057150089382945123710043674866359141242216889254741124762646598254289110441339397731899394735520817723753988225424354848713}{66885843564103432215688508078566334610119874037711946064044975452251542716435140758282334009520258001356717215875351138187144697856} a^{3} + \frac{1141748375395852476632886068456372393134106038892532774594913453740023192493582904276333225625553465663773182894573567151451032967}{16721460891025858053922127019641583652529968509427986516011243863062885679108785189570583502380064500339179303968837784546786174464} a^{2} + \frac{1826605627842433453986214877017501007785638000779358695573362188051608837065048645472115033632693785750878539413897796983050884329}{4180365222756464513480531754910395913132492127356996629002810965765721419777196297392645875595016125084794825992209446136696543616} a + \frac{4059514339167144559546903361021470591856677693764285009081819586266828631493545951311714182496045262816638626839585012301807144031}{16721460891025858053922127019641583652529968509427986516011243863062885679108785189570583502380064500339179303968837784546786174464}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  \( 907073932516000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T48:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3000
The 26 conjugacy class representatives for 1/2[D(5)^3]S(3)
Character table for 1/2[D(5)^3]S(3) is not computed

Intermediate fields

3.1.44.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 R $15$ $15$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $15$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
911Data not computed
1181Data not computed
2570851Data not computed