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Properties

Label 15.1.17283271259...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{12}\cdot 3^{13}\cdot 5^{10}\cdot 11^{13}\cdot 151^{5}$
Root discriminant $561.27$
Ramified primes $2, 3, 5, 11, 151$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-122095500, -80668500, -18672300, 2426400, -17048850, 1635325, 1456855, -338145, -115260, -18615, 1893, 1485, 60, -45, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 45*x^13 + 60*x^12 + 1485*x^11 + 1893*x^10 - 18615*x^9 - 115260*x^8 - 338145*x^7 + 1456855*x^6 + 1635325*x^5 - 17048850*x^4 + 2426400*x^3 - 18672300*x^2 - 80668500*x - 122095500)
 
gp: K = bnfinit(x^15 - 5*x^14 - 45*x^13 + 60*x^12 + 1485*x^11 + 1893*x^10 - 18615*x^9 - 115260*x^8 - 338145*x^7 + 1456855*x^6 + 1635325*x^5 - 17048850*x^4 + 2426400*x^3 - 18672300*x^2 - 80668500*x - 122095500, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 45 x^{13} + 60 x^{12} + 1485 x^{11} + 1893 x^{10} - 18615 x^{9} - 115260 x^{8} - 338145 x^{7} + 1456855 x^{6} + 1635325 x^{5} - 17048850 x^{4} + 2426400 x^{3} - 18672300 x^{2} - 80668500 x - 122095500 \)

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:  \(-172832712591425821508917768538520000000000=-\,2^{12}\cdot 3^{13}\cdot 5^{10}\cdot 11^{13}\cdot 151^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $561.27$
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:  $2, 3, 5, 11, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} + \frac{1}{5} a^{5} - \frac{3}{10} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{330} a^{10} + \frac{1}{165} a^{9} - \frac{1}{30} a^{8} - \frac{137}{330} a^{7} + \frac{131}{330} a^{6} - \frac{25}{66} a^{5} + \frac{4}{11} a^{4} + \frac{7}{22} a^{3} + \frac{3}{11} a - \frac{4}{11}$, $\frac{1}{330} a^{11} - \frac{1}{22} a^{9} + \frac{17}{330} a^{8} + \frac{3}{110} a^{7} + \frac{47}{110} a^{6} + \frac{53}{165} a^{5} - \frac{1}{110} a^{4} + \frac{4}{11} a^{3} + \frac{3}{11} a^{2} + \frac{1}{11} a - \frac{3}{11}$, $\frac{1}{1650} a^{12} + \frac{8}{165} a^{9} + \frac{1}{22} a^{8} - \frac{12}{25} a^{7} - \frac{47}{165} a^{6} - \frac{23}{55} a^{5} + \frac{29}{110} a^{4} - \frac{27}{55} a^{3} - \frac{2}{11} a^{2} + \frac{4}{11} a - \frac{1}{11}$, $\frac{1}{1650} a^{13} + \frac{8}{165} a^{9} - \frac{7}{150} a^{8} + \frac{26}{165} a^{7} + \frac{109}{330} a^{6} - \frac{157}{330} a^{5} + \frac{43}{110} a^{4} + \frac{5}{22} a^{3} + \frac{4}{11} a^{2} - \frac{5}{11} a - \frac{2}{11}$, $\frac{1}{34877103031809019736977248999600788204550} a^{14} - \frac{1218837334767438421782055706104028018}{5812850505301503289496208166600131367425} a^{13} + \frac{1027764851382166510086352937936825777}{3875233670201002192997472111066754244950} a^{12} + \frac{58335635987825029254241307941634629}{77504673404020043859949442221335084899} a^{11} - \frac{503557984183160118731445644217394989}{775046734040200438599494422213350848990} a^{10} + \frac{284603423204044228190893467509893447643}{5812850505301503289496208166600131367425} a^{9} + \frac{725185682286090536711363761027553813527}{11625701010603006578992416333200262734850} a^{8} + \frac{1616043772307476499360243515513982355499}{5812850505301503289496208166600131367425} a^{7} - \frac{6096555021412965221132444349781223983}{70458794003654585327226765655759168090} a^{6} + \frac{275771245309665774327772326881239117729}{3487710303180901973697724899960078820455} a^{5} - \frac{360787208632848519059504365222786773223}{2325140202120601315798483266640052546970} a^{4} + \frac{100373461413687566366869369985970516588}{387523367020100219299747211106675424495} a^{3} - \frac{21046271596354560740731301618027397349}{77504673404020043859949442221335084899} a^{2} - \frac{4229188255819071897677889759172446904}{77504673404020043859949442221335084899} a + \frac{60683874239573599801135356291632894138}{232514020212060131579848326664005254697}$

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:  \( 161101180902933.84 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.24915.1, 5.1.2371842000.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.2.1.1$x^{2} - 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} - 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} - 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} - 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} - 151$$2$$1$$1$$C_2$$[\ ]_{2}$