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Properties

Label 15.1.21005140039...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{12}\cdot 5^{9}\cdot 11^{13}\cdot 3769^{5}$
Root discriminant $568.62$
Ramified primes $2, 5, 11, 3769$
Class number $25$ (GRH)
Class group $[5, 5]$ (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![-2807964928, -3039303680, -419916800, 192419200, -81164320, 18555360, 2732880, -2261440, -366980, 8390, 9591, 2225, -190, -40, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 40*x^13 - 190*x^12 + 2225*x^11 + 9591*x^10 + 8390*x^9 - 366980*x^8 - 2261440*x^7 + 2732880*x^6 + 18555360*x^5 - 81164320*x^4 + 192419200*x^3 - 419916800*x^2 - 3039303680*x - 2807964928)
 
gp: K = bnfinit(x^15 - 5*x^14 - 40*x^13 - 190*x^12 + 2225*x^11 + 9591*x^10 + 8390*x^9 - 366980*x^8 - 2261440*x^7 + 2732880*x^6 + 18555360*x^5 - 81164320*x^4 + 192419200*x^3 - 419916800*x^2 - 3039303680*x - 2807964928, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 40 x^{13} - 190 x^{12} + 2225 x^{11} + 9591 x^{10} + 8390 x^{9} - 366980 x^{8} - 2261440 x^{7} + 2732880 x^{6} + 18555360 x^{5} - 81164320 x^{4} + 192419200 x^{3} - 419916800 x^{2} - 3039303680 x - 2807964928 \)

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:  \(-210051400399105472360464905891352000000000=-\,2^{12}\cdot 5^{9}\cdot 11^{13}\cdot 3769^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $568.62$
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:  $2, 5, 11, 3769$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{176} a^{10} - \frac{1}{176} a^{9} - \frac{1}{88} a^{8} - \frac{7}{88} a^{7} + \frac{13}{176} a^{6} + \frac{1}{16} a^{5} + \frac{9}{44} a^{4} + \frac{21}{44} a^{3} - \frac{3}{11} a^{2} - \frac{5}{11} a + \frac{2}{11}$, $\frac{1}{352} a^{11} - \frac{1}{352} a^{10} + \frac{5}{88} a^{9} - \frac{7}{176} a^{8} - \frac{31}{352} a^{7} + \frac{1}{32} a^{6} - \frac{15}{176} a^{5} + \frac{21}{88} a^{4} + \frac{5}{44} a^{3} - \frac{5}{22} a^{2} - \frac{9}{22} a$, $\frac{1}{1760} a^{12} - \frac{1}{1760} a^{11} - \frac{1}{440} a^{10} - \frac{39}{880} a^{9} + \frac{21}{352} a^{8} - \frac{181}{1760} a^{7} - \frac{39}{880} a^{6} + \frac{87}{440} a^{5} - \frac{37}{220} a^{4} - \frac{31}{220} a^{3} + \frac{26}{55} a^{2} + \frac{1}{11} a - \frac{2}{55}$, $\frac{1}{3520} a^{13} - \frac{1}{3520} a^{12} - \frac{1}{880} a^{11} + \frac{1}{1760} a^{10} + \frac{5}{704} a^{9} + \frac{9}{320} a^{8} + \frac{61}{1760} a^{7} + \frac{17}{880} a^{6} - \frac{23}{110} a^{5} - \frac{1}{440} a^{4} + \frac{8}{55} a^{3} + \frac{5}{11} a^{2} + \frac{9}{55} a - \frac{3}{11}$, $\frac{1}{868757649569027417638341417319457692088877773440} a^{14} + \frac{107164063646212778965279834742606871617691017}{868757649569027417638341417319457692088877773440} a^{13} + \frac{114430252688852615135892565019299936401765967}{434378824784513708819170708659728846044438886720} a^{12} - \frac{85706609749291972655068027379850863110064293}{434378824784513708819170708659728846044438886720} a^{11} - \frac{323943751133506650113128876795513006357804363}{868757649569027417638341417319457692088877773440} a^{10} - \frac{40281713404829231279674761685343929681058170539}{868757649569027417638341417319457692088877773440} a^{9} - \frac{5699853737592182102186045706168449960574545799}{217189412392256854409585354329864423022219443360} a^{8} + \frac{26044377748434491973578355650659402017905700877}{217189412392256854409585354329864423022219443360} a^{7} + \frac{2661837119961038279743262594827414840333418959}{21718941239225685440958535432986442302221944336} a^{6} + \frac{1903001850679832201246832796083181778987401309}{9872246017829857018617516105902928319191792880} a^{5} - \frac{4737025341354610963946633452025682698404550289}{27148676549032106801198169291233052877777430420} a^{4} - \frac{5670342805611450952488714654652633892357573773}{13574338274516053400599084645616526438888715210} a^{3} - \frac{1381187883746952259586403878732924357636288904}{6787169137258026700299542322808263219444357605} a^{2} + \frac{2768511517372151117441024741844015642546889891}{6787169137258026700299542322808263219444357605} a + \frac{2806161481469594019224381252334516720030050642}{6787169137258026700299542322808263219444357605}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  \( 36942965423141.99 \) (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

3.1.41459.1, 5.1.29282000.1

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$11$11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.2$x^{10} - 72171$$10$$1$$9$$C_{10}$$[\ ]_{10}$
3769Data not computed