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Properties

Label 15.1.30590839399...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{12}\cdot 5^{15}\cdot 11^{13}\cdot 19^{5}\cdot 31^{5}$
Root discriminant $583.05$
Ramified primes $2, 5, 11, 19, 31$
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![4649536, 89744320, -15785600, 10719680, -404560, 4659456, -2144000, 856800, -178520, 42250, -10297, 1945, -370, 60, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 60*x^13 - 370*x^12 + 1945*x^11 - 10297*x^10 + 42250*x^9 - 178520*x^8 + 856800*x^7 - 2144000*x^6 + 4659456*x^5 - 404560*x^4 + 10719680*x^3 - 15785600*x^2 + 89744320*x + 4649536)
 
gp: K = bnfinit(x^15 - 5*x^14 + 60*x^13 - 370*x^12 + 1945*x^11 - 10297*x^10 + 42250*x^9 - 178520*x^8 + 856800*x^7 - 2144000*x^6 + 4659456*x^5 - 404560*x^4 + 10719680*x^3 - 15785600*x^2 + 89744320*x + 4649536, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 60 x^{13} - 370 x^{12} + 1945 x^{11} - 10297 x^{10} + 42250 x^{9} - 178520 x^{8} + 856800 x^{7} - 2144000 x^{6} + 4659456 x^{5} - 404560 x^{4} + 10719680 x^{3} - 15785600 x^{2} + 89744320 x + 4649536 \)

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:  \(-305908393993716440346185189875000000000000=-\,2^{12}\cdot 5^{15}\cdot 11^{13}\cdot 19^{5}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $583.05$
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:  $2, 5, 11, 19, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5}$, $\frac{1}{88} a^{10} - \frac{1}{8} a^{9} + \frac{3}{44} a^{8} - \frac{1}{4} a^{7} + \frac{21}{88} a^{6} - \frac{1}{8} a^{5} + \frac{3}{22} a^{4} - \frac{1}{2} a^{3} + \frac{2}{11} a^{2} + \frac{4}{11}$, $\frac{1}{176} a^{11} - \frac{1}{176} a^{10} - \frac{1}{11} a^{9} - \frac{3}{88} a^{8} - \frac{23}{176} a^{7} - \frac{21}{176} a^{6} - \frac{5}{88} a^{5} - \frac{3}{44} a^{4} - \frac{9}{22} a^{3} - \frac{1}{11} a^{2} - \frac{7}{22} a - \frac{2}{11}$, $\frac{1}{176} a^{12} - \frac{1}{176} a^{10} - \frac{1}{8} a^{9} - \frac{21}{176} a^{8} - \frac{1}{4} a^{7} + \frac{41}{176} a^{6} - \frac{1}{8} a^{5} + \frac{5}{44} a^{4} - \frac{1}{2} a^{3} - \frac{5}{11} a^{2} - \frac{1}{2} a - \frac{3}{11}$, $\frac{1}{352} a^{13} - \frac{1}{352} a^{12} - \frac{1}{176} a^{10} - \frac{15}{352} a^{9} + \frac{3}{352} a^{8} - \frac{13}{176} a^{7} - \frac{1}{22} a^{6} + \frac{1}{11} a^{5} - \frac{7}{44} a^{4} - \frac{19}{44} a^{3} - \frac{9}{22} a^{2} + \frac{5}{11} a + \frac{4}{11}$, $\frac{1}{5027237252828086718347244390752543720352} a^{14} + \frac{5649008783486880425122380848016313753}{5027237252828086718347244390752543720352} a^{13} - \frac{2936860560885275957822533326181593509}{1256809313207021679586811097688135930088} a^{12} - \frac{1082028182729905690573952690397174729}{1256809313207021679586811097688135930088} a^{11} - \frac{905982737832658333659982889497503767}{173353008718209886839560151405260128288} a^{10} + \frac{55324288845411194275343476953462996381}{5027237252828086718347244390752543720352} a^{9} + \frac{133365623079190520680539688675456446905}{2513618626414043359173622195376271860176} a^{8} - \frac{25700759322744591296685869605747162097}{228510784219458487197602017761479260016} a^{7} - \frac{303250522742244877513749839556956370665}{1256809313207021679586811097688135930088} a^{6} + \frac{251138009137097420385825243282723822943}{1256809313207021679586811097688135930088} a^{5} + \frac{15680147942967419961013465099251794122}{157101164150877709948351387211016991261} a^{4} + \frac{55108993178064352772729823507992184630}{157101164150877709948351387211016991261} a^{3} + \frac{100620962890047002434128599076564799691}{314202328301755419896702774422033982522} a^{2} + \frac{11718105302785616459445835297829285020}{157101164150877709948351387211016991261} a - \frac{1330280408314774922765465044228041426}{157101164150877709948351387211016991261}$

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:  \( 162220877031922.56 \) (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

3.1.6479.1, 5.1.732050000.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$5$5.15.15.40$x^{15} + 10 x^{14} + 5 x^{13} + 20 x^{12} + 20 x^{11} + 17 x^{10} + 10 x^{9} + 15 x^{8} + 12 x^{5} + 15 x^{3} + 10 x^{2} + 20 x + 17$$5$$3$$15$$F_5\times C_3$$[5/4]_{4}^{3}$
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$