# Properties

 Label 15.3.872411699045053489.1 Degree $15$ Signature $[3, 6]$ Discriminant $7^{8}\cdot 73^{6}$ Root discriminant $15.71$ Ramified primes $7, 73$ Class number $1$ Class group Trivial Galois Group $\GL(2,4)$ (as 15T15)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 10, -5, -16, 23, -19, -10, 35, -17, -6, 17, -10, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 10*x^12 + 17*x^11 - 6*x^10 - 17*x^9 + 35*x^8 - 10*x^7 - 19*x^6 + 23*x^5 - 16*x^4 - 5*x^3 + 10*x^2 - 4*x + 1)
gp: K = bnfinit(x^15 - x^14 - 10*x^12 + 17*x^11 - 6*x^10 - 17*x^9 + 35*x^8 - 10*x^7 - 19*x^6 + 23*x^5 - 16*x^4 - 5*x^3 + 10*x^2 - 4*x + 1, 1)

## Normalizeddefining polynomial

$$x^{15}$$ $$\mathstrut -\mathstrut x^{14}$$ $$\mathstrut -\mathstrut 10 x^{12}$$ $$\mathstrut +\mathstrut 17 x^{11}$$ $$\mathstrut -\mathstrut 6 x^{10}$$ $$\mathstrut -\mathstrut 17 x^{9}$$ $$\mathstrut +\mathstrut 35 x^{8}$$ $$\mathstrut -\mathstrut 10 x^{7}$$ $$\mathstrut -\mathstrut 19 x^{6}$$ $$\mathstrut +\mathstrut 23 x^{5}$$ $$\mathstrut -\mathstrut 16 x^{4}$$ $$\mathstrut -\mathstrut 5 x^{3}$$ $$\mathstrut +\mathstrut 10 x^{2}$$ $$\mathstrut -\mathstrut 4 x$$ $$\mathstrut +\mathstrut 1$$

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

## Invariants

 Degree: $15$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[3, 6]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$872411699045053489=7^{8}\cdot 73^{6}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $15.71$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $7, 73$ magma: PrimeDivisors(Discriminant(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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{81341849} a^{14} + \frac{5756519}{81341849} a^{13} - \frac{13742834}{81341849} a^{12} - \frac{12670213}{81341849} a^{11} - \frac{26847007}{81341849} a^{10} + \frac{31930055}{81341849} a^{9} + \frac{20253206}{81341849} a^{8} - \frac{18845186}{81341849} a^{7} + \frac{5592006}{81341849} a^{6} + \frac{27030294}{81341849} a^{5} - \frac{21772177}{81341849} a^{4} + \frac{36646238}{81341849} a^{3} + \frac{3810440}{81341849} a^{2} - \frac{12958077}{81341849} a - \frac{25572480}{81341849}$

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

## Class group and class number

Trivial Abelian group, order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
 Rank: $8$ 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: $$\frac{39395365}{81341849} a^{14} - \frac{13117226}{81341849} a^{13} - \frac{16728669}{81341849} a^{12} - \frac{404773259}{81341849} a^{11} + \frac{402434737}{81341849} a^{10} + \frac{114234791}{81341849} a^{9} - \frac{648811149}{81341849} a^{8} + \frac{907575426}{81341849} a^{7} + \frac{337070698}{81341849} a^{6} - \frac{704131259}{81341849} a^{5} + \frac{420502621}{81341849} a^{4} - \frac{306048274}{81341849} a^{3} - \frac{430147128}{81341849} a^{2} + \frac{175173206}{81341849} a - \frac{432326}{81341849}$$,  $$\frac{35571825}{81341849} a^{14} - \frac{2853576}{81341849} a^{13} - \frac{6277366}{81341849} a^{12} - \frac{347593716}{81341849} a^{11} + \frac{278422534}{81341849} a^{10} + \frac{90220493}{81341849} a^{9} - \frac{687679107}{81341849} a^{8} + \frac{773244951}{81341849} a^{7} + \frac{271510202}{81341849} a^{6} - \frac{625265901}{81341849} a^{5} + \frac{579071168}{81341849} a^{4} - \frac{169905582}{81341849} a^{3} - \frac{417304791}{81341849} a^{2} + \frac{255312490}{81341849} a - \frac{57894670}{81341849}$$,  $$\frac{81774175}{81341849} a^{14} - \frac{42378810}{81341849} a^{13} - \frac{13117226}{81341849} a^{12} - \frac{834470419}{81341849} a^{11} + \frac{985387716}{81341849} a^{10} - \frac{88210313}{81341849} a^{9} - \frac{1275926184}{81341849} a^{8} + \frac{2213284976}{81341849} a^{7} + \frac{89833676}{81341849} a^{6} - \frac{1216638627}{81341849} a^{5} + \frac{1176674766}{81341849} a^{4} - \frac{887884179}{81341849} a^{3} - \frac{714919149}{81341849} a^{2} + \frac{387594622}{81341849} a - \frac{151923494}{81341849}$$,  $$\frac{22322845}{81341849} a^{14} - \frac{19469269}{81341849} a^{13} + \frac{3423790}{81341849} a^{12} - \frac{237630350}{81341849} a^{11} + \frac{358193140}{81341849} a^{10} - \frac{159165979}{81341849} a^{9} - \frac{206309790}{81341849} a^{8} + \frac{565389392}{81341849} a^{7} - \frac{77211951}{81341849} a^{6} - \frac{203222627}{81341849} a^{5} + \frac{127240341}{81341849} a^{4} - \frac{177337970}{81341849} a^{3} - \frac{42074141}{81341849} a^{2} - \frac{93670581}{81341849} a + \frac{81628480}{81341849}$$,  $$\frac{39395365}{81341849} a^{14} - \frac{13117226}{81341849} a^{13} - \frac{16728669}{81341849} a^{12} - \frac{404773259}{81341849} a^{11} + \frac{402434737}{81341849} a^{10} + \frac{114234791}{81341849} a^{9} - \frac{648811149}{81341849} a^{8} + \frac{907575426}{81341849} a^{7} + \frac{337070698}{81341849} a^{6} - \frac{704131259}{81341849} a^{5} + \frac{420502621}{81341849} a^{4} - \frac{306048274}{81341849} a^{3} - \frac{430147128}{81341849} a^{2} + \frac{256515055}{81341849} a - \frac{432326}{81341849}$$,  $$\frac{17992445}{81341849} a^{14} - \frac{26302329}{81341849} a^{13} - \frac{2522782}{81341849} a^{12} - \frac{176610026}{81341849} a^{11} + \frac{392913445}{81341849} a^{10} - \frac{151213141}{81341849} a^{9} - \frac{362485259}{81341849} a^{8} + \frac{735416052}{81341849} a^{7} - \frac{150205202}{81341849} a^{6} - \frac{616914435}{81341849} a^{5} + \frac{570840372}{81341849} a^{4} - \frac{304852506}{81341849} a^{3} - \frac{170671246}{81341849} a^{2} + \frac{295146814}{81341849} a - \frac{50917365}{81341849}$$,  $$\frac{45420689}{81341849} a^{14} - \frac{17520311}{81341849} a^{13} - \frac{18077073}{81341849} a^{12} - \frac{452337905}{81341849} a^{11} + \frac{500338450}{81341849} a^{10} + \frac{113111301}{81341849} a^{9} - \frac{897508014}{81341849} a^{8} + \frac{1059461920}{81341849} a^{7} + \frac{320312315}{81341849} a^{6} - \frac{816758857}{81341849} a^{5} + \frac{554918117}{81341849} a^{4} - \frac{226134226}{81341849} a^{3} - \frac{356812214}{81341849} a^{2} + \frac{268670549}{81341849} a - \frac{111054653}{81341849}$$,  $$\frac{76966833}{81341849} a^{14} - \frac{43467471}{81341849} a^{13} - \frac{10522590}{81341849} a^{12} - \frac{784025508}{81341849} a^{11} + \frac{976818431}{81341849} a^{10} - \frac{124921354}{81341849} a^{9} - \frac{1204057861}{81341849} a^{8} + \frac{2007014650}{81341849} a^{7} + \frac{72201634}{81341849} a^{6} - \frac{1155112279}{81341849} a^{5} + \frac{1097423097}{81341849} a^{4} - \frac{734297187}{81341849} a^{3} - \frac{537432980}{81341849} a^{2} + \frac{267176286}{81341849} a - \frac{91552239}{81341849}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$1080.93773334$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$C_3\times A_5$ (as 15T15):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A non-solvable group of order 180 The 15 conjugacy class representatives for $\GL(2,4)$ Character table for $\GL(2,4)$

## Intermediate fields

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

## Sibling fields

 Degree 15 sibling: data not computed Degree 18 sibling: data not computed Degree 30 sibling: data not computed Degree 36 sibling: data not computed Degree 45 sibling: data not computed Arithmetically equvalently siblings: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $15$ $15$ $15$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $15$ $15$

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
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3} 7.3.2.3x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3} 7373.3.0.1x^{3} - x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
73.12.6.1$x^{12} + 3890170 x^{6} - 2073071593 x^{2} + 3783355657225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$