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Properties

Label 15.9.50913071707...0000.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{18}\cdot 5^{6}\cdot 7^{14}\cdot 43^{4}\cdot 20509^{2}\cdot 356999^{2}$
Root discriminant $1515.15$
Ramified primes $2, 5, 7, 43, 20509, 356999$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T95

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1232896000, -2157568000, -2003456000, 4110937600, 409167360, -1876640640, 486419584, 67412352, -33288192, 3949848, -598528, 61040, -3024, 378, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 378*x^13 - 3024*x^12 + 61040*x^11 - 598528*x^10 + 3949848*x^9 - 33288192*x^8 + 67412352*x^7 + 486419584*x^6 - 1876640640*x^5 + 409167360*x^4 + 4110937600*x^3 - 2003456000*x^2 - 2157568000*x + 1232896000)
 
gp: K = bnfinit(x^15 + 378*x^13 - 3024*x^12 + 61040*x^11 - 598528*x^10 + 3949848*x^9 - 33288192*x^8 + 67412352*x^7 + 486419584*x^6 - 1876640640*x^5 + 409167360*x^4 + 4110937600*x^3 - 2003456000*x^2 - 2157568000*x + 1232896000, 1)
 

Normalized defining polynomial

\( x^{15} + 378 x^{13} - 3024 x^{12} + 61040 x^{11} - 598528 x^{10} + 3949848 x^{9} - 33288192 x^{8} + 67412352 x^{7} + 486419584 x^{6} - 1876640640 x^{5} + 409167360 x^{4} + 4110937600 x^{3} - 2003456000 x^{2} - 2157568000 x + 1232896000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-509130717075076439975600141140915908481024000000=-\,2^{18}\cdot 5^{6}\cdot 7^{14}\cdot 43^{4}\cdot 20509^{2}\cdot 356999^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1515.15$
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:  $2, 5, 7, 43, 20509, 356999$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a$, $\frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{9} - \frac{1}{32} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{128} a^{10} - \frac{1}{64} a^{8} - \frac{1}{16} a^{6} - \frac{1}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{1280} a^{11} - \frac{1}{640} a^{9} + \frac{1}{80} a^{8} - \frac{1}{32} a^{7} + \frac{1}{40} a^{6} + \frac{11}{160} a^{5} + \frac{1}{10} a^{4} - \frac{9}{40} a^{3} - \frac{1}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{2560} a^{12} - \frac{1}{1280} a^{10} + \frac{1}{160} a^{9} - \frac{1}{64} a^{8} + \frac{1}{80} a^{7} + \frac{11}{320} a^{6} - \frac{3}{40} a^{5} - \frac{9}{80} a^{4} + \frac{3}{20} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{51200} a^{13} + \frac{9}{25600} a^{11} + \frac{11}{3200} a^{10} + \frac{1}{320} a^{9} + \frac{1}{100} a^{8} + \frac{131}{6400} a^{7} - \frac{1}{100} a^{6} + \frac{23}{800} a^{5} - \frac{7}{400} a^{4} + \frac{1}{80} a^{3} - \frac{3}{20} a^{2} + \frac{1}{4} a$, $\frac{1}{270867655774953715774331561653884896854972211200} a^{14} + \frac{297436253392007003447953830735653876470971}{67716913943738428943582890413471224213743052800} a^{13} - \frac{23066858649061299439710932297799935530487131}{135433827887476857887165780826942448427486105600} a^{12} + \frac{4623067240762876092739857078318296637223561}{33858456971869214471791445206735612106871526400} a^{11} + \frac{46804507592571668606850287836635440459692569}{16929228485934607235895722603367806053435763200} a^{10} + \frac{7613972973640228404737784139339444048212299}{2116153560741825904486965325420975756679470400} a^{9} + \frac{87467300251611086597454027494945158887696307}{33858456971869214471791445206735612106871526400} a^{8} + \frac{41205889066448933227742675714761297132966621}{1692922848593460723589572260336780605343576320} a^{7} + \frac{41119320586815030402318062114943486520721829}{1058076780370912952243482662710487878339735200} a^{6} - \frac{83407969007274757936052699789132264858294021}{2116153560741825904486965325420975756679470400} a^{5} + \frac{121259349805424830637906185294023111779117097}{2116153560741825904486965325420975756679470400} a^{4} + \frac{1539568247251191204817978343920474680566883}{52903839018545647612174133135524393916986760} a^{3} + \frac{6053282866973487369506007753170979916892203}{105807678037091295224348266271048787833973520} a^{2} + \frac{1214998405923146260997642716943779612125973}{5290383901854564761217413313552439391698676} a - \frac{33186998130689990788440283809855778506974}{1322595975463641190304353328388109847924669}$

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:  $11$
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:  \( 2962071577270000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T95:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1296000
The 53 conjugacy class representatives for [A(5)^3:2]3 are not computed
Character table for [A(5)^3:2]3 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 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ $15$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ $15$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]^{3}$
2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.6$x^{6} + 4 x^{2} + 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
43Data not computed
20509Data not computed
356999Data not computed