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Properties

Label 15.15.6425324665...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{22}\cdot 3^{16}\cdot 5^{10}\cdot 7^{8}\cdot 43^{6}$
Root discriminant $331.53$
Ramified primes $2, 3, 5, 7, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T84

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-178921805, 877767145, -1297426135, 416494315, 435539901, -241308801, -43933115, 28369855, 1967915, -1366411, -43169, 30665, 425, -305, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 305*x^13 + 425*x^12 + 30665*x^11 - 43169*x^10 - 1366411*x^9 + 1967915*x^8 + 28369855*x^7 - 43933115*x^6 - 241308801*x^5 + 435539901*x^4 + 416494315*x^3 - 1297426135*x^2 + 877767145*x - 178921805)
 
gp: K = bnfinit(x^15 - x^14 - 305*x^13 + 425*x^12 + 30665*x^11 - 43169*x^10 - 1366411*x^9 + 1967915*x^8 + 28369855*x^7 - 43933115*x^6 - 241308801*x^5 + 435539901*x^4 + 416494315*x^3 - 1297426135*x^2 + 877767145*x - 178921805, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 305 x^{13} + 425 x^{12} + 30665 x^{11} - 43169 x^{10} - 1366411 x^{9} + 1967915 x^{8} + 28369855 x^{7} - 43933115 x^{6} - 241308801 x^{5} + 435539901 x^{4} + 416494315 x^{3} - 1297426135 x^{2} + 877767145 x - 178921805 \)

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:  \(64253246659742539130123427840000000000=2^{22}\cdot 3^{16}\cdot 5^{10}\cdot 7^{8}\cdot 43^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $331.53$
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:  $2, 3, 5, 7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4}$, $\frac{1}{10} a^{10} + \frac{1}{10} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{9} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{100} a^{12} - \frac{1}{50} a^{10} - \frac{1}{25} a^{9} + \frac{7}{20} a^{8} - \frac{8}{25} a^{7} + \frac{3}{10} a^{6} + \frac{1}{25} a^{5} + \frac{13}{100} a^{4} + \frac{2}{5} a^{3} - \frac{1}{2} a^{2} - \frac{3}{20}$, $\frac{1}{100} a^{13} - \frac{1}{50} a^{11} - \frac{1}{25} a^{10} - \frac{1}{20} a^{9} + \frac{7}{25} a^{8} - \frac{1}{2} a^{7} + \frac{11}{25} a^{6} - \frac{7}{100} a^{5} - \frac{1}{5} a^{4} - \frac{1}{2} a^{3} - \frac{3}{20} a$, $\frac{1}{2313532666730984881468635147922826249362485830275000} a^{14} + \frac{3345959599992584832168907549404782036595469641983}{1156766333365492440734317573961413124681242915137500} a^{13} - \frac{5194941378084241457039680847498503251309744725683}{2313532666730984881468635147922826249362485830275000} a^{12} + \frac{13540602841077569127421169598529319029471742061308}{289191583341373110183579393490353281170310728784375} a^{11} + \frac{46314240256529121341787575509443313740372110060853}{2313532666730984881468635147922826249362485830275000} a^{10} + \frac{22396855618553952496163873644676309548927510316591}{1156766333365492440734317573961413124681242915137500} a^{9} - \frac{478737219351595029860030877606252060578904781601417}{2313532666730984881468635147922826249362485830275000} a^{8} + \frac{133834832648696887846941561668068423918721812983419}{578383166682746220367158786980706562340621457568750} a^{7} + \frac{339339009570556954107824044487386823734080628623547}{2313532666730984881468635147922826249362485830275000} a^{6} - \frac{352917363345472821074199904905342177920910276790583}{1156766333365492440734317573961413124681242915137500} a^{5} - \frac{170203326368119825803319012514375865704956664556823}{2313532666730984881468635147922826249362485830275000} a^{4} - \frac{9573311922442933381918047481017130349806866067461}{57838316668274622036715878698070656234062145756875} a^{3} + \frac{134226389192702942600276728898052985801578502330067}{462706533346196976293727029584565249872497166055000} a^{2} + \frac{45018194897334955620738490199042899409752152406531}{231353266673098488146863514792282624936248583027500} a + \frac{211101810066808667896012759423280342225185587433383}{462706533346196976293727029584565249872497166055000}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1284426000550000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T84:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 77760
The 39 conjugacy class representatives for 1/2[S(3)^5]F(5)
Character table for 1/2[S(3)^5]F(5) is not computed

Intermediate fields

5.5.3698000.1

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

Sibling fields

Degree 30 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 R R R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $15$ $15$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ $15$ R ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.18.9$x^{10} - 2 x^{9} - 6 x^{8} - 6$$10$$1$$18$$(C_2^4 : C_5):C_4$$[14/5, 14/5, 14/5, 14/5]_{5}^{4}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.12.16.6$x^{12} + 120 x^{11} - 117 x^{10} - 57 x^{9} + 36 x^{8} + 54 x^{7} - 18 x^{6} + 81 x^{5} + 81$$3$$4$$16$12T46$[2, 2]^{8}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$