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Properties

Label 15.3.10106301800...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{23}\cdot 3^{13}\cdot 5^{17}\cdot 17^{13}$
Root discriminant $541.55$
Ramified primes $2, 3, 5, 17$
Class number $100$ (GRH)
Class group $[5, 20]$ (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![-164654741844, -67359711780, -15893213040, -3522839040, -557303040, -190329732, -21172140, -4697580, 154410, -134070, 8033, -2395, 500, -20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 20*x^13 + 500*x^12 - 2395*x^11 + 8033*x^10 - 134070*x^9 + 154410*x^8 - 4697580*x^7 - 21172140*x^6 - 190329732*x^5 - 557303040*x^4 - 3522839040*x^3 - 15893213040*x^2 - 67359711780*x - 164654741844)
 
gp: K = bnfinit(x^15 - 5*x^14 - 20*x^13 + 500*x^12 - 2395*x^11 + 8033*x^10 - 134070*x^9 + 154410*x^8 - 4697580*x^7 - 21172140*x^6 - 190329732*x^5 - 557303040*x^4 - 3522839040*x^3 - 15893213040*x^2 - 67359711780*x - 164654741844, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 20 x^{13} + 500 x^{12} - 2395 x^{11} + 8033 x^{10} - 134070 x^{9} + 154410 x^{8} - 4697580 x^{7} - 21172140 x^{6} - 190329732 x^{5} - 557303040 x^{4} - 3522839040 x^{3} - 15893213040 x^{2} - 67359711780 x - 164654741844 \)

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:  \(101063018004202830052166400000000000000000=2^{23}\cdot 3^{13}\cdot 5^{17}\cdot 17^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $541.55$
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:  $2, 3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{18} a^{12} + \frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{1}{18} a^{8} - \frac{1}{18} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{18} a^{13} + \frac{1}{18} a^{10} - \frac{1}{6} a^{9} - \frac{1}{9} a^{7} + \frac{1}{6} a^{6}$, $\frac{1}{496593123016765421506133504163457415851655282481076775691399036} a^{14} - \frac{5358173761271495311759968357438345524221840619151395867192239}{496593123016765421506133504163457415851655282481076775691399036} a^{13} - \frac{3349446789941127294647423221275666373648059210555291157529551}{124148280754191355376533376040864353962913820620269193922849759} a^{12} - \frac{892224507478435331263044059458638044907981765320047291291627}{27588506834264745639229639120192078658425293471170931982855502} a^{11} - \frac{12606247185548161717864649048820394092300063808482960252113327}{496593123016765421506133504163457415851655282481076775691399036} a^{10} - \frac{39187030474760761641222967481152501162655879464558775786222567}{496593123016765421506133504163457415851655282481076775691399036} a^{9} + \frac{669038792320807126999483613033577662594243401602402178239761}{124148280754191355376533376040864353962913820620269193922849759} a^{8} + \frac{1822153120920379057499150730409886137054622675040854318260477}{41382760251397118458844458680288117987637940206756397974283253} a^{7} - \frac{520416920748908191228109024565403452656615844992595430572692}{41382760251397118458844458680288117987637940206756397974283253} a^{6} - \frac{30140002069964493460908746981990066977152758077563909863981617}{82765520502794236917688917360576235975275880413512795948566506} a^{5} + \frac{1020180377812690332872802565335119396702327644265775734974519}{82765520502794236917688917360576235975275880413512795948566506} a^{4} - \frac{2776741174673506884797669939280473143152992505045804314483683}{13794253417132372819614819560096039329212646735585465991427751} a^{3} + \frac{1449920958933066357976207953393929545613679177273182651547627}{13794253417132372819614819560096039329212646735585465991427751} a^{2} - \frac{5785446524657423292450108582023483069210843145553220388523899}{13794253417132372819614819560096039329212646735585465991427751} a - \frac{2211696397900107766692574120093615488857083804431488848332443}{13794253417132372819614819560096039329212646735585465991427751}$

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

Class group and class number

$C_{5}\times C_{20}$, 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:  $8$
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:  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:  \( 22441515369714.94 \) (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

3.3.10200.1, 5.1.338260050000.3

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{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$$[5/4]_{12}^{2}$
$17$17.5.4.1$x^{5} - 17$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
17.10.9.2$x^{10} + 51$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$