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Properties

Label 15.1.21210997669...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 3^{12}\cdot 5^{10}\cdot 19^{13}\cdot 41^{5}$
Root discriminant $901.79$
Ramified primes $2, 3, 5, 19, 41$
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![69750425364, 93251552652, 69424749540, -1739418204, -1851693840, 982709201, 45129354, -9952441, 2742312, 90422, 4644, 7120, -552, 115, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 115*x^13 - 552*x^12 + 7120*x^11 + 4644*x^10 + 90422*x^9 + 2742312*x^8 - 9952441*x^7 + 45129354*x^6 + 982709201*x^5 - 1851693840*x^4 - 1739418204*x^3 + 69424749540*x^2 + 93251552652*x + 69750425364)
 
gp: K = bnfinit(x^15 - 6*x^14 + 115*x^13 - 552*x^12 + 7120*x^11 + 4644*x^10 + 90422*x^9 + 2742312*x^8 - 9952441*x^7 + 45129354*x^6 + 982709201*x^5 - 1851693840*x^4 - 1739418204*x^3 + 69424749540*x^2 + 93251552652*x + 69750425364, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} + 115 x^{13} - 552 x^{12} + 7120 x^{11} + 4644 x^{10} + 90422 x^{9} + 2742312 x^{8} - 9952441 x^{7} + 45129354 x^{6} + 982709201 x^{5} - 1851693840 x^{4} - 1739418204 x^{3} + 69424749540 x^{2} + 93251552652 x + 69750425364 \)

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:  \(-212109976697868913424682627876372480000000000=-\,2^{23}\cdot 3^{12}\cdot 5^{10}\cdot 19^{13}\cdot 41^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $901.79$
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:  $2, 3, 5, 19, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{12} + \frac{3}{10} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{8} + \frac{2}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{5} a^{4} + \frac{3}{10} a^{3} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{530663451236354960763078570157999558186770106178551587436544827319138110} a^{14} + \frac{14968753228709654513828891047927237810817032668781042648119679304963147}{530663451236354960763078570157999558186770106178551587436544827319138110} a^{13} - \frac{35812756574660732486634987168629239053384378453726336854914137442563275}{106132690247270992152615714031599911637354021235710317487308965463827622} a^{12} + \frac{46131822268553862902680437145707462184230433837334431413510831621227039}{106132690247270992152615714031599911637354021235710317487308965463827622} a^{11} - \frac{17673811743125344793370403733725345264190202330014747218478751420677019}{265331725618177480381539285078999779093385053089275793718272413659569055} a^{10} + \frac{111849882139535513867379231465316389526078836683119418143360747457291546}{265331725618177480381539285078999779093385053089275793718272413659569055} a^{9} - \frac{5932306680869525352412313270122212995412258112721029532813379640676826}{265331725618177480381539285078999779093385053089275793718272413659569055} a^{8} + \frac{94337672363350335399829783589423578071305619837731907411112146160926732}{265331725618177480381539285078999779093385053089275793718272413659569055} a^{7} + \frac{99895028866010137217434332717666155569306435182211379583410374667213291}{530663451236354960763078570157999558186770106178551587436544827319138110} a^{6} + \frac{259736181363760416625498205244755944618222044488596012645243981503034483}{530663451236354960763078570157999558186770106178551587436544827319138110} a^{5} + \frac{31562077076974015116103194927389441419569226097042641603156185155833219}{106132690247270992152615714031599911637354021235710317487308965463827622} a^{4} + \frac{77632675177825554642039222649283542449436659242842349078265765035019577}{530663451236354960763078570157999558186770106178551587436544827319138110} a^{3} - \frac{111579405345528226601965494407859944182160721662815660198559034658386353}{265331725618177480381539285078999779093385053089275793718272413659569055} a^{2} - \frac{7144583750756240804627890099800739342236729866782708166802478150879959}{265331725618177480381539285078999779093385053089275793718272413659569055} a - \frac{121723992471881579937396581438111868456936151101985564409121627101890408}{265331725618177480381539285078999779093385053089275793718272413659569055}$

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:  \( 9379097783625176.0 \) (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

3.1.31160.1, 5.1.21112002000.4

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 R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ R ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$$[\ ]_{5}^{4}$
2.10.19.1$x^{10} - 2$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.10.9.2$x^{10} + 76$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$