↑

Properties

Label 15.1.21919782836...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 3^{12}\cdot 5^{9}\cdot 31^{13}\cdot 61^{5}$
Root discriminant $775.16$
Ramified primes $2, 3, 5, 31, 61$
Class number $75$ (GRH)
Class group $[5, 15]$ (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![963981822, -216139110, 237235950, -35275635, 35139510, 6608476, 2275555, 1776615, -171900, 161325, -14202, 6825, -465, 135, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 135*x^13 - 465*x^12 + 6825*x^11 - 14202*x^10 + 161325*x^9 - 171900*x^8 + 1776615*x^7 + 2275555*x^6 + 6608476*x^5 + 35139510*x^4 - 35275635*x^3 + 237235950*x^2 - 216139110*x + 963981822)
 
gp: K = bnfinit(x^15 - 5*x^14 + 135*x^13 - 465*x^12 + 6825*x^11 - 14202*x^10 + 161325*x^9 - 171900*x^8 + 1776615*x^7 + 2275555*x^6 + 6608476*x^5 + 35139510*x^4 - 35275635*x^3 + 237235950*x^2 - 216139110*x + 963981822, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 135 x^{13} - 465 x^{12} + 6825 x^{11} - 14202 x^{10} + 161325 x^{9} - 171900 x^{8} + 1776615 x^{7} + 2275555 x^{6} + 6608476 x^{5} + 35139510 x^{4} - 35275635 x^{3} + 237235950 x^{2} - 216139110 x + 963981822 \)

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:  \(-21919782836729069869914926177537862000000000=-\,2^{10}\cdot 3^{12}\cdot 5^{9}\cdot 31^{13}\cdot 61^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $775.16$
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:  $2, 3, 5, 31, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{93} a^{10} - \frac{40}{93} a^{9} + \frac{22}{93} a^{8} - \frac{38}{93} a^{7} - \frac{4}{93} a^{6} + \frac{1}{3} a^{5} - \frac{15}{31} a^{4} + \frac{5}{31} a^{3} + \frac{7}{31} a^{2} + \frac{9}{31} a + \frac{12}{31}$, $\frac{1}{279} a^{11} + \frac{1}{93} a^{9} + \frac{5}{279} a^{8} - \frac{4}{31} a^{7} - \frac{4}{31} a^{6} + \frac{79}{279} a^{5} - \frac{2}{31} a^{4} - \frac{10}{93} a^{3} - \frac{7}{31} a^{2} + \frac{46}{93}$, $\frac{1}{1395} a^{12} + \frac{2}{1395} a^{11} + \frac{1}{465} a^{10} - \frac{268}{1395} a^{9} + \frac{532}{1395} a^{8} - \frac{74}{155} a^{7} + \frac{7}{1395} a^{6} - \frac{139}{1395} a^{5} + \frac{164}{465} a^{4} - \frac{227}{465} a^{3} - \frac{9}{31} a^{2} + \frac{139}{465} a + \frac{92}{465}$, $\frac{1}{1395} a^{13} - \frac{1}{1395} a^{11} - \frac{4}{1395} a^{10} + \frac{11}{465} a^{9} + \frac{5}{279} a^{8} - \frac{551}{1395} a^{7} + \frac{18}{155} a^{6} - \frac{125}{279} a^{5} + \frac{3}{31} a^{4} - \frac{191}{465} a^{3} - \frac{26}{465} a^{2} - \frac{27}{155} a - \frac{199}{465}$, $\frac{1}{1972355333240546064887695128847754729443046745} a^{14} - \frac{704799413040151670811061092438176189058458}{1972355333240546064887695128847754729443046745} a^{13} - \frac{200485327521985869428491457777982924248947}{657451777746848688295898376282584909814348915} a^{12} - \frac{140491527919014209240156697944275369218191}{1972355333240546064887695128847754729443046745} a^{11} - \frac{1926960857229417418589800800900326209972895}{394471066648109212977539025769550945888609349} a^{10} + \frac{18584534898797118096257444808372982893766282}{657451777746848688295898376282584909814348915} a^{9} + \frac{855542479696544537589289914011032460830196954}{1972355333240546064887695128847754729443046745} a^{8} + \frac{167522237264446058549895350209223832399256487}{394471066648109212977539025769550945888609349} a^{7} - \frac{22208562993742725093579340661465302963234139}{219150592582282896098632792094194969938116305} a^{6} + \frac{27513822558973361389146826525864346809441945}{394471066648109212977539025769550945888609349} a^{5} - \frac{99212682250585605032340096583530331958668902}{219150592582282896098632792094194969938116305} a^{4} + \frac{264611816725784357596650108784052742029425397}{657451777746848688295898376282584909814348915} a^{3} - \frac{94615198607901637275961837483720570003628453}{657451777746848688295898376282584909814348915} a^{2} - \frac{52361641251639871118663352428595158493676017}{219150592582282896098632792094194969938116305} a + \frac{143506793071094938705334495560319545787643647}{657451777746848688295898376282584909814348915}$

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

Class group and class number

$C_{5}\times C_{15}$, which has order $75$ (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:  \( 185624234491811.5 \) (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

3.1.7564.1, 5.1.9350650125.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$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}$
$31$31.5.4.3$x^{5} - 1519$$5$$1$$4$$C_5$$[\ ]_{5}$
31.10.9.2$x^{10} - 1519$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$