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Properties

Label 15.15.2368379989...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 3^{5}\cdot 5^{25}\cdot 11^{13}\cdot 41^{13}$
Root discriminant $6682.88$
Ramified primes $2, 3, 5, 11, 41$
Class number $1375$ (GRH)
Class group $[5, 5, 55]$ (GRH)
Galois group $S_3 \times C_5$ (as 15T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1257096395450004601701, 316610688507343605265, 1700941616424112590, -5640651315561778120, -457716480374582000, 8195553887899983, 2079870790777010, 44485161759555, -2216106307965, -76578988200, 891723965, 45081960, -124025, -11275, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 11275*x^13 - 124025*x^12 + 45081960*x^11 + 891723965*x^10 - 76578988200*x^9 - 2216106307965*x^8 + 44485161759555*x^7 + 2079870790777010*x^6 + 8195553887899983*x^5 - 457716480374582000*x^4 - 5640651315561778120*x^3 + 1700941616424112590*x^2 + 316610688507343605265*x + 1257096395450004601701)
 
gp: K = bnfinit(x^15 - 11275*x^13 - 124025*x^12 + 45081960*x^11 + 891723965*x^10 - 76578988200*x^9 - 2216106307965*x^8 + 44485161759555*x^7 + 2079870790777010*x^6 + 8195553887899983*x^5 - 457716480374582000*x^4 - 5640651315561778120*x^3 + 1700941616424112590*x^2 + 316610688507343605265*x + 1257096395450004601701, 1)
 

Normalized defining polynomial

\( x^{15} - 11275 x^{13} - 124025 x^{12} + 45081960 x^{11} + 891723965 x^{10} - 76578988200 x^{9} - 2216106307965 x^{8} + 44485161759555 x^{7} + 2079870790777010 x^{6} + 8195553887899983 x^{5} - 457716480374582000 x^{4} - 5640651315561778120 x^{3} + 1700941616424112590 x^{2} + 316610688507343605265 x + 1257096395450004601701 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2368379989452191012928309708195070432434082031250000000000=2^{10}\cdot 3^{5}\cdot 5^{25}\cdot 11^{13}\cdot 41^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $6682.88$
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:  $2, 3, 5, 11, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{33} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{99} a^{6} - \frac{1}{99} a^{5} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{2}{9} a^{2} - \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{99} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{297} a^{8} - \frac{1}{297} a^{7} + \frac{1}{99} a^{5} + \frac{4}{9} a^{3} - \frac{2}{27} a^{2} + \frac{5}{27} a - \frac{1}{9}$, $\frac{1}{891} a^{9} + \frac{1}{891} a^{8} - \frac{2}{891} a^{7} + \frac{1}{297} a^{6} - \frac{1}{297} a^{5} - \frac{2}{27} a^{4} + \frac{22}{81} a^{3} - \frac{35}{81} a^{2} - \frac{29}{81} a - \frac{11}{27}$, $\frac{1}{401841} a^{10} + \frac{2}{891} a^{7} + \frac{1}{297} a^{6} - \frac{1}{99} a^{5} + \frac{1}{81} a^{4} - \frac{4}{27} a^{3} + \frac{2}{9} a^{2} + \frac{20}{81} a - \frac{10}{27}$, $\frac{1}{401841} a^{11} - \frac{1}{891} a^{8} - \frac{1}{297} a^{7} - \frac{7}{891} a^{5} + \frac{2}{27} a^{4} - \frac{1}{9} a^{3} - \frac{37}{81} a^{2} + \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{3616569} a^{12} + \frac{1}{1205523} a^{11} - \frac{1}{3616569} a^{10} + \frac{2}{8019} a^{9} - \frac{20}{8019} a^{7} + \frac{35}{8019} a^{6} - \frac{1}{891} a^{5} + \frac{8}{729} a^{4} + \frac{113}{729} a^{3} + \frac{100}{243} a^{2} + \frac{277}{729} a - \frac{8}{243}$, $\frac{1}{32549121} a^{13} - \frac{4}{32549121} a^{12} - \frac{4}{32549121} a^{11} + \frac{2}{3616569} a^{10} + \frac{40}{72171} a^{9} + \frac{16}{72171} a^{8} + \frac{175}{72171} a^{7} + \frac{313}{72171} a^{6} + \frac{673}{72171} a^{5} - \frac{80}{2187} a^{4} - \frac{1976}{6561} a^{3} + \frac{2263}{6561} a^{2} + \frac{602}{6561} a + \frac{326}{2187}$, $\frac{1}{6961188146034467578507678257806307242738542888463295205232414327362874850413865311326045011} a^{14} - \frac{47095561692852559482949803614585173107348068285480408680301760219391309005615540638}{6961188146034467578507678257806307242738542888463295205232414327362874850413865311326045011} a^{13} - \frac{280440065729300771345218720989011143659098254920228091360426995044999307954084919009}{2320396048678155859502559419268769080912847629487765068410804775787624950137955103775348337} a^{12} + \frac{6421001331437397772825818592353034708296327438963049451852686517526106931732371599682}{6961188146034467578507678257806307242738542888463295205232414327362874850413865311326045011} a^{11} + \frac{608424053702074310705333172375039239706329820604697878739362922622445307255123223203}{632835286003133416227970750709664294794412989860299564112037666123897713673987755575095001} a^{10} - \frac{997674267262853873414929423630783025112969568150490501099038884916057178675877684880}{5145002325228726961203014233411904835726934876913004586276728992877217184341363866464187} a^{9} + \frac{776528680149756968920497649085844663245043431401568829359095947489246710828179046638}{1715000775076242320401004744470634945242311625637668195425576330959072394780454622154729} a^{8} - \frac{10187108143124978224812649767802956286303008163737688254550114149279901157952300014623}{5145002325228726961203014233411904835726934876913004586276728992877217184341363866464187} a^{7} + \frac{54620449709935957417094930821484195624093665739522575947938271162771042701220804119}{467727484111702451018455839401082257793357716083000416934248090261565198576487624224017} a^{6} + \frac{133901936340258991629569835730985013055765802158337977998571865683570055815870365410252}{15435006975686180883609042700235714507180804630739013758830186978631651553024091599392561} a^{5} - \frac{24653175506193754732525100005353002908520326898421688081666325260368540102435845783474}{1403182452335107353055367518203246773380073148249001250802744270784695595729462872672051} a^{4} - \frac{216109863404929038922459505644714362571471532257997400444345106319141359199286281482991}{467727484111702451018455839401082257793357716083000416934248090261565198576487624224017} a^{3} - \frac{409189364505354960241086660213987096621072462892697254237602399799478946071561938600061}{1403182452335107353055367518203246773380073148249001250802744270784695595729462872672051} a^{2} - \frac{691748021247226002585062908387179999676020942514416458266033310706742880328648602172458}{1403182452335107353055367518203246773380073148249001250802744270784695595729462872672051} a - \frac{10376295138364685186964483582640777160774976058123808187792225020266523915699345204858}{467727484111702451018455839401082257793357716083000416934248090261565198576487624224017}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{55}$, which has order $1375$ (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:  $14$
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:  \( 42948975476702135000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times S_3$ (as 15T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 15 conjugacy class representatives for $S_3 \times C_5$
Character table for $S_3 \times C_5$

Intermediate fields

3.3.27060.1, 5.5.16160924531640625.6

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 R R R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ R $15$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$2.15.10.1$x^{15} + 8 x^{6} + 32$$3$$5$$10$$S_3 \times C_5$$[\ ]_{3}^{10}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.5.8.4$x^{5} - 5 x^{4} + 55$$5$$1$$8$$C_5$$[2]$
5.10.17.27$x^{10} - 10 x^{8} + 10$$10$$1$$17$$C_{10}$$[2]_{2}$
$11$11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.9$x^{10} + 297$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$41$41.5.4.3$x^{5} - 1476$$5$$1$$4$$C_5$$[\ ]_{5}$
41.10.9.2$x^{10} - 1476$$10$$1$$9$$C_{10}$$[\ ]_{10}$