↑

Properties

Label 15.5.46431713006...9375.2
Degree $15$
Signature $[5, 5]$
Discriminant $-\,3^{6}\cdot 5^{24}\cdot 11^{12}\cdot 23^{7}$
Root discriminant $599.50$
Ramified primes $3, 5, 11, 23$
Class number $125$ (GRH)
Class group $[5, 5, 5]$ (GRH)
Galois group 15T40

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![368982332191957, 79865978259705, -69620528681495, -16147355822675, 168219899100, 100194176808, 10086284725, -1077527385, -70074125, 4277350, 1106633, 12650, -4620, -330, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 330*x^13 - 4620*x^12 + 12650*x^11 + 1106633*x^10 + 4277350*x^9 - 70074125*x^8 - 1077527385*x^7 + 10086284725*x^6 + 100194176808*x^5 + 168219899100*x^4 - 16147355822675*x^3 - 69620528681495*x^2 + 79865978259705*x + 368982332191957)
 
gp: K = bnfinit(x^15 - 330*x^13 - 4620*x^12 + 12650*x^11 + 1106633*x^10 + 4277350*x^9 - 70074125*x^8 - 1077527385*x^7 + 10086284725*x^6 + 100194176808*x^5 + 168219899100*x^4 - 16147355822675*x^3 - 69620528681495*x^2 + 79865978259705*x + 368982332191957, 1)
 

Normalized defining polynomial

\( x^{15} - 330 x^{13} - 4620 x^{12} + 12650 x^{11} + 1106633 x^{10} + 4277350 x^{9} - 70074125 x^{8} - 1077527385 x^{7} + 10086284725 x^{6} + 100194176808 x^{5} + 168219899100 x^{4} - 16147355822675 x^{3} - 69620528681495 x^{2} + 79865978259705 x + 368982332191957 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-464317130069216763190044343471527099609375=-\,3^{6}\cdot 5^{24}\cdot 11^{12}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $599.50$
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:  $3, 5, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{11} a^{6}$, $\frac{1}{11} a^{7}$, $\frac{1}{11} a^{8}$, $\frac{1}{11} a^{9}$, $\frac{1}{121} a^{10}$, $\frac{1}{121} a^{11}$, $\frac{1}{121} a^{12}$, $\frac{1}{25531} a^{13} + \frac{93}{25531} a^{12} + \frac{72}{25531} a^{11} - \frac{20}{25531} a^{10} - \frac{96}{2321} a^{9} + \frac{57}{2321} a^{8} + \frac{43}{2321} a^{7} - \frac{5}{211} a^{6} - \frac{5}{211} a^{5} - \frac{99}{211} a^{4} + \frac{72}{211} a^{3} - \frac{1}{211} a^{2} + \frac{39}{211} a + \frac{57}{211}$, $\frac{1}{57915918383823518593184749845664328982772801049641618859127380876269973809867870651} a^{14} + \frac{108395328648284827071580724791388855233018386896243094133253879086022281879792}{57915918383823518593184749845664328982772801049641618859127380876269973809867870651} a^{13} - \frac{234538328882045303987896640788967389248652139176427308141891609827211675352950888}{57915918383823518593184749845664328982772801049641618859127380876269973809867870651} a^{12} - \frac{210654819040684566197974287463483204887246002619473541089783901886121977272551703}{57915918383823518593184749845664328982772801049641618859127380876269973809867870651} a^{11} - \frac{101436198644773495074785331777252224864114029017129863377271373489823683616801137}{57915918383823518593184749845664328982772801049641618859127380876269973809867870651} a^{10} + \frac{98170677217240364425248328654747328098177947613842934928745637759974052429258790}{5265083489438501690289522713242211725706618277240147169011580079660906709987988241} a^{9} + \frac{218013321083278768953842975532594940023727989184842576971905036188012818820536305}{5265083489438501690289522713242211725706618277240147169011580079660906709987988241} a^{8} - \frac{2717361317131789329480150011513335599226206767241870798647188723220657475166101}{752154784205500241469931816177458817958088325320021024144511439951558101426855463} a^{7} - \frac{41350878411898631181369678360158469548607722519396623275963280913789468616565390}{5265083489438501690289522713242211725706618277240147169011580079660906709987988241} a^{6} - \frac{113067697503582250737937841356659071599003803843563729037121749677790538477659115}{5265083489438501690289522713242211725706618277240147169011580079660906709987988241} a^{5} + \frac{17658382060507475633654075270524916211702370578513213264568161696781314607562716}{68377707655045476497266528743405347087098938665456456740410130904687100129714133} a^{4} - \frac{98028845495506527048992134603646126052424062870806889136541872760727402132862351}{478643953585318335480865701203837429609692570658195197182870916332809700907998931} a^{3} - \frac{209075473135280264774754939235120199510708114999441832523953081011023132859843589}{478643953585318335480865701203837429609692570658195197182870916332809700907998931} a^{2} - \frac{12622216601243081552184888327853274038492087999904664847580517394091189813028147}{478643953585318335480865701203837429609692570658195197182870916332809700907998931} a + \frac{13513143686273809628430248518059435924120274576274480594176671584719561631531393}{478643953585318335480865701203837429609692570658195197182870916332809700907998931}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{5}$, which has order $125$ (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:  $9$
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:  \( 15044332102100 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T40:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1500
The 40 conjugacy class representatives for [5^3:2]S(3)
Character table for [5^3:2]S(3) is not computed

Intermediate fields

3.1.23.1

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

Sibling fields

Degree 15 siblings: data not computed
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.10.16.10$x^{10} + 40 x^{9} + 10 x^{8} + 70 x^{7} + 15 x^{6} + 120 x^{5} + 5 x^{4} - 45 x^{3} - 20 x^{2} + 90 x + 7$$5$$2$$16$$C_{10}$$[2]^{2}$
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.3$x^{10} - 11 x^{5} + 847$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$