↑

Properties

Label 15.1.10362578395...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 5^{17}\cdot 61^{13}$
Root discriminant $632.46$
Ramified primes $2, 5, 61$
Class number $100$ (GRH)
Class group $[10, 10]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1744422601, -600143435, -592258115, -387843005, -136700405, 23655453, 13204145, -21445, -703515, -147465, 6647, 5665, 185, -105, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 105*x^13 + 185*x^12 + 5665*x^11 + 6647*x^10 - 147465*x^9 - 703515*x^8 - 21445*x^7 + 13204145*x^6 + 23655453*x^5 - 136700405*x^4 - 387843005*x^3 - 592258115*x^2 - 600143435*x - 1744422601)
 
gp: K = bnfinit(x^15 - 5*x^14 - 105*x^13 + 185*x^12 + 5665*x^11 + 6647*x^10 - 147465*x^9 - 703515*x^8 - 21445*x^7 + 13204145*x^6 + 23655453*x^5 - 136700405*x^4 - 387843005*x^3 - 592258115*x^2 - 600143435*x - 1744422601, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 105 x^{13} + 185 x^{12} + 5665 x^{11} + 6647 x^{10} - 147465 x^{9} - 703515 x^{8} - 21445 x^{7} + 13204145 x^{6} + 23655453 x^{5} - 136700405 x^{4} - 387843005 x^{3} - 592258115 x^{2} - 600143435 x - 1744422601 \)

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:  \(-1036257839565777636208518400000000000000000=-\,2^{23}\cdot 5^{17}\cdot 61^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $632.46$
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:  $2, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{20} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a + \frac{9}{20}$, $\frac{1}{20} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{5} a + \frac{1}{4}$, $\frac{1}{20} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{5} a^{2} + \frac{1}{4}$, $\frac{1}{40} a^{8} - \frac{1}{4} a^{4} + \frac{1}{10} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{200} a^{9} + \frac{1}{200} a^{8} - \frac{1}{50} a^{7} - \frac{1}{50} a^{6} - \frac{1}{50} a^{5} + \frac{1}{50} a^{4} + \frac{1}{50} a^{3} + \frac{21}{50} a^{2} + \frac{59}{200} a + \frac{59}{200}$, $\frac{1}{24400} a^{10} + \frac{3}{6100} a^{9} - \frac{163}{24400} a^{8} - \frac{1}{3050} a^{7} + \frac{101}{12200} a^{6} - \frac{17}{1220} a^{5} - \frac{801}{12200} a^{4} - \frac{72}{1525} a^{3} + \frac{7293}{24400} a^{2} + \frac{363}{1525} a + \frac{10749}{24400}$, $\frac{1}{24400} a^{11} + \frac{59}{24400} a^{9} - \frac{63}{12200} a^{8} + \frac{27}{12200} a^{7} - \frac{71}{3050} a^{6} - \frac{103}{12200} a^{5} + \frac{309}{6100} a^{4} - \frac{5479}{24400} a^{3} - \frac{423}{3050} a^{2} - \frac{4413}{24400} a + \frac{1813}{12200}$, $\frac{1}{244000} a^{12} - \frac{1}{122000} a^{11} - \frac{1}{61000} a^{10} - \frac{39}{24400} a^{9} - \frac{203}{48800} a^{8} + \frac{567}{61000} a^{7} - \frac{407}{30500} a^{6} - \frac{943}{61000} a^{5} + \frac{5663}{48800} a^{4} - \frac{281}{24400} a^{3} - \frac{21861}{61000} a^{2} + \frac{40129}{122000} a - \frac{20149}{244000}$, $\frac{1}{244000} a^{13} + \frac{1}{122000} a^{11} + \frac{1}{122000} a^{10} - \frac{13}{48800} a^{9} - \frac{1391}{122000} a^{8} + \frac{53}{12200} a^{7} - \frac{1481}{61000} a^{6} - \frac{5049}{244000} a^{5} - \frac{33}{200} a^{4} - \frac{24947}{122000} a^{3} + \frac{5349}{24400} a^{2} + \frac{31897}{244000} a - \frac{37239}{122000}$, $\frac{1}{1441521332576955549348423906404000} a^{14} + \frac{61168351403819696392483821}{1441521332576955549348423906404000} a^{13} - \frac{135627626527256430317378641}{72076066628847777467421195320200} a^{12} + \frac{321245616813769893220150977}{360380333144238887337105976601000} a^{11} + \frac{105393238311695855427123145}{11532170660615644394787391251232} a^{10} - \frac{1346660309771656226803443423347}{1441521332576955549348423906404000} a^{9} + \frac{417815971884463320746147008847}{360380333144238887337105976601000} a^{8} - \frac{727071181186443516542233225101}{36038033314423888733710597660100} a^{7} + \frac{11062132074780033386855408541459}{1441521332576955549348423906404000} a^{6} - \frac{7102403996857478344720456342981}{288304266515391109869684781280800} a^{5} - \frac{57008394819316568056034662102101}{360380333144238887337105976601000} a^{4} + \frac{75476064691419189005823284921}{2418659953988180451926885749000} a^{3} + \frac{28051237777985236705444398201067}{288304266515391109869684781280800} a^{2} + \frac{426887121984588060895182698775183}{1441521332576955549348423906404000} a - \frac{2608258566463459334257287873581}{72076066628847777467421195320200}$

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

Class group and class number

$C_{10}\times C_{10}$, which has order $100$ (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:  \( 26746181066439.71 \) (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

3.1.12200.1, 5.1.692292050000.2

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.19.49$x^{10} - 6$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$5$5.15.17.3$x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10$$15$$1$$17$$F_5 \times S_3$$[5/4]_{12}^{2}$
$61$61.5.4.3$x^{5} - 244$$5$$1$$4$$C_5$$[\ ]_{5}$
61.10.9.9$x^{10} + 7808$$10$$1$$9$$C_{10}$$[\ ]_{10}$