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Properties

Label 15.1.26283114663...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 5^{27}\cdot 19^{13}$
Root discriminant $672.94$
Ramified primes $2, 5, 19$
Class number $400$ (GRH)
Class group $[2, 10, 20]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![46251096148, 1630311940, -1694606600, 648860840, -62922700, 40460928, -1263140, 505680, 180490, -36230, 10769, 25, 20, 0, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 20*x^12 + 25*x^11 + 10769*x^10 - 36230*x^9 + 180490*x^8 + 505680*x^7 - 1263140*x^6 + 40460928*x^5 - 62922700*x^4 + 648860840*x^3 - 1694606600*x^2 + 1630311940*x + 46251096148)
 
gp: K = bnfinit(x^15 - 5*x^14 + 20*x^12 + 25*x^11 + 10769*x^10 - 36230*x^9 + 180490*x^8 + 505680*x^7 - 1263140*x^6 + 40460928*x^5 - 62922700*x^4 + 648860840*x^3 - 1694606600*x^2 + 1630311940*x + 46251096148, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 20 x^{12} + 25 x^{11} + 10769 x^{10} - 36230 x^{9} + 180490 x^{8} + 505680 x^{7} - 1263140 x^{6} + 40460928 x^{5} - 62922700 x^{4} + 648860840 x^{3} - 1694606600 x^{2} + 1630311940 x + 46251096148 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2628311466391066187500000000000000000000000=-\,2^{23}\cdot 5^{27}\cdot 19^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $672.94$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{19} a^{6} + \frac{9}{19} a^{5} - \frac{2}{19} a^{3} + \frac{7}{19} a - \frac{7}{19}$, $\frac{1}{19} a^{7} - \frac{5}{19} a^{5} - \frac{2}{19} a^{4} - \frac{1}{19} a^{3} + \frac{7}{19} a^{2} + \frac{6}{19} a + \frac{6}{19}$, $\frac{1}{19} a^{8} + \frac{5}{19} a^{5} - \frac{1}{19} a^{4} - \frac{3}{19} a^{3} + \frac{6}{19} a^{2} + \frac{3}{19} a + \frac{3}{19}$, $\frac{1}{19} a^{9} - \frac{8}{19} a^{5} - \frac{3}{19} a^{4} - \frac{3}{19} a^{3} + \frac{3}{19} a^{2} + \frac{6}{19} a - \frac{3}{19}$, $\frac{1}{722} a^{10} + \frac{15}{722} a^{9} - \frac{6}{361} a^{8} - \frac{2}{361} a^{7} + \frac{1}{722} a^{6} + \frac{89}{722} a^{5} + \frac{176}{361} a^{4} - \frac{105}{361} a^{3} + \frac{61}{361} a^{2} + \frac{26}{361} a + \frac{144}{361}$, $\frac{1}{722} a^{11} - \frac{9}{722} a^{9} - \frac{7}{361} a^{8} - \frac{15}{722} a^{7} - \frac{1}{361} a^{6} + \frac{271}{722} a^{5} - \frac{28}{361} a^{4} - \frac{112}{361} a^{3} + \frac{61}{361} a^{2} + \frac{20}{361} a + \frac{139}{361}$, $\frac{1}{722} a^{12} + \frac{7}{722} a^{9} - \frac{9}{722} a^{8} + \frac{7}{361} a^{6} - \frac{357}{722} a^{5} + \frac{104}{361} a^{4} + \frac{85}{361} a^{3} + \frac{151}{361} a^{2} + \frac{107}{361} a + \frac{156}{361}$, $\frac{1}{13718} a^{13} + \frac{3}{13718} a^{12} + \frac{4}{6859} a^{11} - \frac{1}{6859} a^{10} + \frac{337}{13718} a^{9} - \frac{145}{13718} a^{8} + \frac{98}{6859} a^{7} - \frac{94}{6859} a^{6} + \frac{24}{6859} a^{5} - \frac{974}{6859} a^{4} + \frac{1348}{6859} a^{3} + \frac{2247}{6859} a^{2} - \frac{3340}{6859} a - \frac{2946}{6859}$, $\frac{1}{15076062440602894794551941367075839849349603644} a^{14} + \frac{200956078046508453910445927288433358551179}{15076062440602894794551941367075839849349603644} a^{13} + \frac{3728054414133214841162843222686874027320877}{7538031220301447397275970683537919924674801822} a^{12} + \frac{72161907804594449708334604100227432421794}{198369242639511773612525544303629471701968469} a^{11} + \frac{4285968976959595805282845803658786017908435}{15076062440602894794551941367075839849349603644} a^{10} + \frac{204024899696702931651536924756825550646999655}{15076062440602894794551941367075839849349603644} a^{9} + \frac{67730626487551428022706084118568335007626275}{7538031220301447397275970683537919924674801822} a^{8} - \frac{37027700780450614991521820999917328598350333}{7538031220301447397275970683537919924674801822} a^{7} - \frac{180731523209755028326702676571629050380885821}{7538031220301447397275970683537919924674801822} a^{6} - \frac{1233898089160419280676383876211224563277107745}{7538031220301447397275970683537919924674801822} a^{5} - \frac{3022692590431073415268849647667206687059812953}{7538031220301447397275970683537919924674801822} a^{4} - \frac{432916126552077459073748183236463038769924}{198369242639511773612525544303629471701968469} a^{3} - \frac{1324559884123184692597651730752485208215324186}{3769015610150723698637985341768959962337400911} a^{2} - \frac{20043864718683981108917797005278080910249140}{198369242639511773612525544303629471701968469} a + \frac{895877573068481636117579690552264109164139767}{3769015610150723698637985341768959962337400911}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{10}\times C_{20}$, which has order $400$ (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:  $7$
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:  \( 9655767972734.52 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.152.1, 5.1.4072531250000.23

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.19.17$x^{10} - 2 x^{4} + 4 x^{2} - 10$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$5$5.5.9.5$x^{5} + 105$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.18.5$x^{10} + 10 x^{8} + 40 x^{6} + 85 x^{5} + 80 x^{4} - 75 x^{3} + 80 x^{2} + 75 x + 57$$5$$2$$18$$F_{5}\times C_2$$[9/4]_{4}^{2}$
$19$19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.10.9.1$x^{10} - 19$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$