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Properties

Label 15.15.6373627540...9169.1
Degree $15$
Signature $[15, 0]$
Discriminant $7^{10}\cdot 41^{12}$
Root discriminant $71.39$
Ramified primes $7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{15}$ (as 15T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![547, 17047, -27115, -57764, 92292, 44261, -98196, 10106, 29167, -7356, -3368, 1100, 150, -59, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 59*x^13 + 150*x^12 + 1100*x^11 - 3368*x^10 - 7356*x^9 + 29167*x^8 + 10106*x^7 - 98196*x^6 + 44261*x^5 + 92292*x^4 - 57764*x^3 - 27115*x^2 + 17047*x + 547)
 
gp: K = bnfinit(x^15 - 2*x^14 - 59*x^13 + 150*x^12 + 1100*x^11 - 3368*x^10 - 7356*x^9 + 29167*x^8 + 10106*x^7 - 98196*x^6 + 44261*x^5 + 92292*x^4 - 57764*x^3 - 27115*x^2 + 17047*x + 547, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 59 x^{13} + 150 x^{12} + 1100 x^{11} - 3368 x^{10} - 7356 x^{9} + 29167 x^{8} + 10106 x^{7} - 98196 x^{6} + 44261 x^{5} + 92292 x^{4} - 57764 x^{3} - 27115 x^{2} + 17047 x + 547 \)

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:  \(6373627540905023204410809169=7^{10}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.39$
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:  $7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(287=7\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{287}(256,·)$, $\chi_{287}(1,·)$, $\chi_{287}(18,·)$, $\chi_{287}(100,·)$, $\chi_{287}(37,·)$, $\chi_{287}(141,·)$, $\chi_{287}(78,·)$, $\chi_{287}(16,·)$, $\chi_{287}(242,·)$, $\chi_{287}(51,·)$, $\chi_{287}(247,·)$, $\chi_{287}(57,·)$, $\chi_{287}(92,·)$, $\chi_{287}(221,·)$, $\chi_{287}(165,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{3} a^{7} - \frac{2}{9} a^{6} + \frac{2}{9} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{747} a^{13} - \frac{13}{747} a^{12} + \frac{59}{747} a^{11} - \frac{91}{747} a^{10} + \frac{5}{83} a^{9} - \frac{59}{249} a^{8} + \frac{106}{747} a^{7} + \frac{305}{747} a^{6} - \frac{44}{249} a^{5} + \frac{5}{83} a^{4} - \frac{109}{249} a^{3} - \frac{118}{249} a^{2} + \frac{370}{747} a + \frac{36}{83}$, $\frac{1}{196844169779395513799373} a^{14} - \frac{82421716258735761455}{196844169779395513799373} a^{13} + \frac{8642779212866528606392}{196844169779395513799373} a^{12} - \frac{30954461635109389728595}{196844169779395513799373} a^{11} + \frac{2202572987140352097355}{21871574419932834866597} a^{10} - \frac{31102841920537390883827}{196844169779395513799373} a^{9} + \frac{91895393203835695678642}{196844169779395513799373} a^{8} - \frac{96882774218170367244722}{196844169779395513799373} a^{7} + \frac{17128848708806409345890}{196844169779395513799373} a^{6} + \frac{21580896147683521189526}{196844169779395513799373} a^{5} - \frac{340557134634558280306}{65614723259798504599791} a^{4} + \frac{5547690257249055617021}{21871574419932834866597} a^{3} + \frac{18137007720882433334896}{196844169779395513799373} a^{2} - \frac{64418494976667915180943}{196844169779395513799373} a - \frac{64116065030109393896588}{196844169779395513799373}$

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:  \( 873491470.835 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{15}$ (as 15T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 15
The 15 conjugacy class representatives for $C_{15}$
Character table for $C_{15}$

Intermediate fields

\(\Q(\zeta_{7})^+\), 5.5.2825761.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $15$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ $15$ R $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ $15$ $15$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ $15$ $15$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ $15$ $15$ $15$

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
7Data not computed
$41$41.5.4.1$x^{5} - 41$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.1$x^{5} - 41$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.1$x^{5} - 41$$5$$1$$4$$C_5$$[\ ]_{5}$