↑

Properties

Label 15.7.62285142048...0529.1
Degree $15$
Signature $[7, 4]$
Discriminant $7^{10}\cdot 3848209^{2}\cdot 3858721^{2}$
Root discriminant $208.75$
Ramified primes $7, 3848209, 3858721$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T50

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, -5490, -2362248, 240215638, -407256, -57180412, 103900, 5695781, -7001, -303942, 196, 9164, -2, -148, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 148*x^13 - 2*x^12 + 9164*x^11 + 196*x^10 - 303942*x^9 - 7001*x^8 + 5695781*x^7 + 103900*x^6 - 57180412*x^5 - 407256*x^4 + 240215638*x^3 - 2362248*x^2 - 5490*x + 27)
 
gp: K = bnfinit(x^15 - 148*x^13 - 2*x^12 + 9164*x^11 + 196*x^10 - 303942*x^9 - 7001*x^8 + 5695781*x^7 + 103900*x^6 - 57180412*x^5 - 407256*x^4 + 240215638*x^3 - 2362248*x^2 - 5490*x + 27, 1)
 

Normalized defining polynomial

\( x^{15} - 148 x^{13} - 2 x^{12} + 9164 x^{11} + 196 x^{10} - 303942 x^{9} - 7001 x^{8} + 5695781 x^{7} + 103900 x^{6} - 57180412 x^{5} - 407256 x^{4} + 240215638 x^{3} - 2362248 x^{2} - 5490 x + 27 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(62285142048708605347358118488040529=7^{10}\cdot 3848209^{2}\cdot 3858721^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $208.75$
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:  $7, 3848209, 3858721$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{48} a^{8} + \frac{1}{48} a^{7} + \frac{1}{6} a^{6} + \frac{11}{24} a^{5} - \frac{5}{12} a^{4} - \frac{11}{24} a^{3} + \frac{5}{24} a^{2} + \frac{1}{48} a + \frac{1}{16}$, $\frac{1}{192} a^{9} - \frac{1}{96} a^{8} + \frac{5}{192} a^{7} + \frac{47}{96} a^{6} + \frac{5}{96} a^{5} + \frac{19}{96} a^{4} - \frac{5}{48} a^{3} + \frac{67}{192} a^{2} + \frac{1}{4} a + \frac{13}{64}$, $\frac{1}{2304} a^{10} - \frac{1}{2304} a^{9} + \frac{1}{768} a^{8} - \frac{95}{768} a^{7} + \frac{13}{288} a^{6} + \frac{7}{16} a^{5} - \frac{31}{128} a^{4} + \frac{239}{2304} a^{3} - \frac{269}{2304} a^{2} - \frac{227}{768} a + \frac{111}{256}$, $\frac{1}{9216} a^{11} + \frac{1}{4608} a^{9} - \frac{5}{512} a^{8} + \frac{11}{9216} a^{7} - \frac{533}{1152} a^{6} + \frac{11}{1536} a^{5} + \frac{449}{9216} a^{4} - \frac{709}{1536} a^{3} - \frac{1819}{4608} a^{2} + \frac{469}{1536} a - \frac{337}{1024}$, $\frac{1}{110592} a^{12} - \frac{1}{36864} a^{11} + \frac{1}{55296} a^{10} - \frac{1}{384} a^{9} + \frac{665}{110592} a^{8} + \frac{3959}{110592} a^{7} - \frac{5473}{18432} a^{6} - \frac{20101}{110592} a^{5} - \frac{4505}{12288} a^{4} - \frac{10583}{27648} a^{3} + \frac{33}{128} a^{2} + \frac{2821}{12288} a + \frac{401}{4096}$, $\frac{1}{5308416} a^{13} + \frac{11}{2654208} a^{12} + \frac{215}{5308416} a^{11} - \frac{119}{2654208} a^{10} + \frac{3257}{5308416} a^{9} - \frac{1819}{663552} a^{8} - \frac{649543}{5308416} a^{7} + \frac{751013}{5308416} a^{6} - \frac{446351}{2654208} a^{5} - \frac{1866677}{5308416} a^{4} - \frac{52247}{1327104} a^{3} + \frac{130469}{589824} a^{2} + \frac{6335}{36864} a - \frac{81719}{196608}$, $\frac{1}{254803968} a^{14} - \frac{13}{254803968} a^{13} + \frac{7}{84934656} a^{12} - \frac{275}{254803968} a^{11} + \frac{12739}{254803968} a^{10} - \frac{18379}{28311552} a^{9} + \frac{615467}{84934656} a^{8} - \frac{12005107}{127401984} a^{7} + \frac{63024595}{254803968} a^{6} - \frac{18267977}{84934656} a^{5} - \frac{109141213}{254803968} a^{4} - \frac{110395295}{254803968} a^{3} - \frac{12030367}{28311552} a^{2} - \frac{13737013}{28311552} a + \frac{2903749}{9437184}$

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:  $10$
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:  \( 1308217127750 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T50:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3000
The 32 conjugacy class representatives for [D(5)^3]3=D(5)wr3
Character table for [D(5)^3]3=D(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ $15$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{11}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
3848209Data not computed
3858721Data not computed