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Properties

Label 15.15.1433933689...0912.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 7^{10}\cdot 41^{3}\cdot 113^{3}\cdot 167^{2}\cdot 21139^{2}$
Root discriminant $257.30$
Ramified primes $2, 7, 41, 113, 167, 21139$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T75

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2696142673, -923236804, 2616279449, 919513141, -777961186, -316063440, 75956192, 39991098, -1249703, -1754426, -24133, 34805, 593, -310, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 310*x^13 + 593*x^12 + 34805*x^11 - 24133*x^10 - 1754426*x^9 - 1249703*x^8 + 39991098*x^7 + 75956192*x^6 - 316063440*x^5 - 777961186*x^4 + 919513141*x^3 + 2616279449*x^2 - 923236804*x - 2696142673)
 
gp: K = bnfinit(x^15 - 3*x^14 - 310*x^13 + 593*x^12 + 34805*x^11 - 24133*x^10 - 1754426*x^9 - 1249703*x^8 + 39991098*x^7 + 75956192*x^6 - 316063440*x^5 - 777961186*x^4 + 919513141*x^3 + 2616279449*x^2 - 923236804*x - 2696142673, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 310 x^{13} + 593 x^{12} + 34805 x^{11} - 24133 x^{10} - 1754426 x^{9} - 1249703 x^{8} + 39991098 x^{7} + 75956192 x^{6} - 316063440 x^{5} - 777961186 x^{4} + 919513141 x^{3} + 2616279449 x^{2} - 923236804 x - 2696142673 \)

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:  \(1433933689966325663458039537302310912=2^{12}\cdot 7^{10}\cdot 41^{3}\cdot 113^{3}\cdot 167^{2}\cdot 21139^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $257.30$
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:  $2, 7, 41, 113, 167, 21139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{3}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{477969689981769334465133742811833704935139580400567544} a^{14} - \frac{3174011567464727460654262884553088624431414402043391}{119492422495442333616283435702958426233784895100141886} a^{13} - \frac{13460987474254628495668013656926699085624886392157455}{477969689981769334465133742811833704935139580400567544} a^{12} + \frac{6487674642011265923532264694333408270410541510130903}{238984844990884667232566871405916852467569790200283772} a^{11} - \frac{7142532354106622867729137829920962825767152779658809}{477969689981769334465133742811833704935139580400567544} a^{10} + \frac{5845998080801249596056638480244122670469595154743331}{59746211247721166808141717851479213116892447550070943} a^{9} - \frac{94499847089447275649268248661550277108289917817953165}{477969689981769334465133742811833704935139580400567544} a^{8} - \frac{2108715495722710590827655007832798224301091715818433}{119492422495442333616283435702958426233784895100141886} a^{7} - \frac{7416901357386101568829726516760196417625343823474048}{59746211247721166808141717851479213116892447550070943} a^{6} - \frac{8705057063488211915399769226038350077106973531572225}{59746211247721166808141717851479213116892447550070943} a^{5} - \frac{30147076712313712595198684334631380742335912744868293}{119492422495442333616283435702958426233784895100141886} a^{4} - \frac{47099257111653333078994758916998108781384843142088809}{238984844990884667232566871405916852467569790200283772} a^{3} - \frac{63902550400441102061707302978466163581927530442095477}{477969689981769334465133742811833704935139580400567544} a^{2} + \frac{26844356065248763228285617340412098976221547982193831}{238984844990884667232566871405916852467569790200283772} a - \frac{4444242452574997278901722103599552173664742886530651}{477969689981769334465133742811833704935139580400567544}$

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:  \( 35696771522200 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T75:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24000
The 55 conjugacy class representatives for [F(5)^3]3=F(5)wr3 are not computed
Character table for [F(5)^3]3=F(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 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 $15$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{7}$ $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
$113$$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.4.3.3$x^{4} + 339$$4$$1$$3$$C_4$$[\ ]_{4}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$167$$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
167.4.2.2$x^{4} - 167 x^{2} + 139445$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
167.4.0.1$x^{4} - x + 60$$1$$4$$0$$C_4$$[\ ]^{4}$
167.5.0.1$x^{5} - x + 3$$1$$5$$0$$C_5$$[\ ]^{5}$
21139Data not computed