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Properties

Label 15.11.9876983227...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{18}\cdot 3^{25}\cdot 5^{6}\cdot 7^{2}\cdot 11^{5}\cdot 47^{4}\cdot 1153^{2}\cdot 74561^{2}$
Root discriminant $2509.81$
Ramified primes $2, 3, 5, 7, 11, 47, 1153, 74561$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T100

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3008000000000, -1015200000000, 688080000000, 170839125000, -14833200000, -11065807500, -702248000, 413865150, 28595400, -9177561, -351336, 109824, 1918, -609, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 - 609*x^13 + 1918*x^12 + 109824*x^11 - 351336*x^10 - 9177561*x^9 + 28595400*x^8 + 413865150*x^7 - 702248000*x^6 - 11065807500*x^5 - 14833200000*x^4 + 170839125000*x^3 + 688080000000*x^2 - 1015200000000*x - 3008000000000)
 
gp: K = bnfinit(x^15 - 6*x^14 - 609*x^13 + 1918*x^12 + 109824*x^11 - 351336*x^10 - 9177561*x^9 + 28595400*x^8 + 413865150*x^7 - 702248000*x^6 - 11065807500*x^5 - 14833200000*x^4 + 170839125000*x^3 + 688080000000*x^2 - 1015200000000*x - 3008000000000, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} - 609 x^{13} + 1918 x^{12} + 109824 x^{11} - 351336 x^{10} - 9177561 x^{9} + 28595400 x^{8} + 413865150 x^{7} - 702248000 x^{6} - 11065807500 x^{5} - 14833200000 x^{4} + 170839125000 x^{3} + 688080000000 x^{2} - 1015200000000 x - 3008000000000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(987698322701449818222428064455485979759874048000000=2^{18}\cdot 3^{25}\cdot 5^{6}\cdot 7^{2}\cdot 11^{5}\cdot 47^{4}\cdot 1153^{2}\cdot 74561^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2509.81$
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:  $2, 3, 5, 7, 11, 47, 1153, 74561$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{7} + \frac{2}{25} a^{6} + \frac{2}{25} a^{5} - \frac{6}{25} a^{4} - \frac{9}{25} a^{3} + \frac{12}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{50} a^{8} + \frac{3}{50} a^{6} - \frac{6}{25} a^{4} + \frac{11}{50} a^{2}$, $\frac{1}{500} a^{9} + \frac{1}{125} a^{8} - \frac{9}{500} a^{7} + \frac{12}{125} a^{6} + \frac{6}{125} a^{5} + \frac{61}{125} a^{4} + \frac{39}{500} a^{3} + \frac{11}{50} a^{2} + \frac{3}{10} a$, $\frac{1}{5000} a^{10} + \frac{1}{1250} a^{9} + \frac{31}{5000} a^{8} + \frac{7}{1250} a^{7} + \frac{1}{1250} a^{6} + \frac{38}{625} a^{5} - \frac{1021}{5000} a^{4} + \frac{199}{500} a^{3} + \frac{1}{20} a^{2} - \frac{1}{10} a$, $\frac{1}{250000} a^{11} + \frac{1}{62500} a^{10} - \frac{19}{250000} a^{9} - \frac{259}{31250} a^{8} - \frac{173}{125000} a^{7} + \frac{4901}{62500} a^{6} - \frac{4821}{250000} a^{5} - \frac{9461}{25000} a^{4} + \frac{207}{500} a^{3} - \frac{9}{100} a^{2} + \frac{39}{100} a + \frac{2}{5}$, $\frac{1}{2500000} a^{12} + \frac{1}{625000} a^{11} - \frac{19}{2500000} a^{10} + \frac{107}{625000} a^{9} - \frac{2673}{1250000} a^{8} - \frac{1028}{78125} a^{7} - \frac{239821}{2500000} a^{6} - \frac{19461}{250000} a^{5} - \frac{353}{5000} a^{4} + \frac{229}{500} a^{3} - \frac{361}{1000} a^{2} - \frac{21}{100} a$, $\frac{1}{50000000} a^{13} - \frac{3}{25000000} a^{12} - \frac{9}{50000000} a^{11} - \frac{841}{25000000} a^{10} - \frac{5697}{6250000} a^{9} - \frac{57567}{6250000} a^{8} + \frac{326839}{50000000} a^{7} - \frac{22131}{250000} a^{6} - \frac{2629}{1000000} a^{5} - \frac{71}{6250} a^{4} + \frac{4989}{20000} a^{3} + \frac{1}{25} a^{2} + \frac{101}{400} a - \frac{2}{5}$, $\frac{1}{16211450382887807519788166244912603604000000000} a^{14} + \frac{29162309846631119397691025532718716517}{8105725191443903759894083122456301802000000000} a^{13} - \frac{636508791890404687538153402959430371649}{16211450382887807519788166244912603604000000000} a^{12} + \frac{3360629080377902574455806130754190006679}{8105725191443903759894083122456301802000000000} a^{11} - \frac{20282755438601795489026518106951732978133}{506607824465243984993380195153518862625000000} a^{10} - \frac{524444908608752330554726014848529166777597}{2026431297860975939973520780614075450500000000} a^{9} + \frac{159747863632396232032147340208716755865381799}{16211450382887807519788166244912603604000000000} a^{8} + \frac{7731251587569948199218690825913297269995619}{405286259572195187994704156122815090100000000} a^{7} + \frac{1219097950075614839238191937050048886400119}{324229007657756150395763324898252072080000000} a^{6} + \frac{218606255487846607335808446602022104912593}{4052862595721951879947041561228150901000000} a^{5} + \frac{2444744693200431004684647240596463158945517}{6484580153155123007915266497965041441600000} a^{4} + \frac{24162981538377855636092826375704950305127}{81057251914439037598940831224563018020000} a^{3} - \frac{13355602970105782499390261860004570813831}{129691603063102460158305329959300828832000} a^{2} - \frac{465729021058869341244284333786631348427}{1621145038288780751978816624491260360400} a - \frac{6126134742083775731165050233458882994}{20264312978609759399735207806140754505}$

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:  $12$
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:  \( 588721260702000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T100:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 79 conjugacy class representatives for [1/2.S(5)^3]S(3) are not computed
Character table for [1/2.S(5)^3]S(3) is not computed

Intermediate fields

3.3.2673.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 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 R R R R R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47Data not computed
1153Data not computed
74561Data not computed