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Properties

Label 15.1.60306175898...0487.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,23^{7}\cdot 191^{8}$
Root discriminant $71.13$
Ramified primes $23, 191$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $(C_5^2 : C_3):C_2$ (as 15T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21568, -34528, 10000, 42632, 11076, -19022, -55285, 33564, 4820, -3181, -47, 69, 47, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 4*x^13 + 47*x^12 + 69*x^11 - 47*x^10 - 3181*x^9 + 4820*x^8 + 33564*x^7 - 55285*x^6 - 19022*x^5 + 11076*x^4 + 42632*x^3 + 10000*x^2 - 34528*x - 21568)
 
gp: K = bnfinit(x^15 - x^14 - 4*x^13 + 47*x^12 + 69*x^11 - 47*x^10 - 3181*x^9 + 4820*x^8 + 33564*x^7 - 55285*x^6 - 19022*x^5 + 11076*x^4 + 42632*x^3 + 10000*x^2 - 34528*x - 21568, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 4 x^{13} + 47 x^{12} + 69 x^{11} - 47 x^{10} - 3181 x^{9} + 4820 x^{8} + 33564 x^{7} - 55285 x^{6} - 19022 x^{5} + 11076 x^{4} + 42632 x^{3} + 10000 x^{2} - 34528 x - 21568 \)

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:  \(-6030617589845994121689520487=-\,23^{7}\cdot 191^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.13$
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:  $23, 191$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{4} a^{7} + \frac{5}{12} a^{6} + \frac{1}{4} a^{5} - \frac{1}{3} a^{3} - \frac{1}{12} a^{2} - \frac{1}{6} a$, $\frac{1}{6696} a^{12} + \frac{79}{6696} a^{11} + \frac{67}{837} a^{10} - \frac{115}{2232} a^{9} - \frac{479}{6696} a^{8} + \frac{1129}{6696} a^{7} + \frac{2915}{6696} a^{6} + \frac{23}{558} a^{5} - \frac{5}{54} a^{4} - \frac{1681}{6696} a^{3} - \frac{109}{1116} a^{2} - \frac{67}{837} a - \frac{98}{837}$, $\frac{1}{13392} a^{13} - \frac{1}{13392} a^{12} - \frac{17}{1116} a^{11} - \frac{817}{13392} a^{10} - \frac{779}{13392} a^{9} - \frac{2959}{13392} a^{8} + \frac{625}{4464} a^{7} - \frac{199}{3348} a^{6} + \frac{1579}{3348} a^{5} - \frac{767}{4464} a^{4} + \frac{2185}{6696} a^{3} - \frac{1283}{3348} a^{2} - \frac{13}{558} a + \frac{14}{837}$, $\frac{1}{4446374650270322252232153888} a^{14} - \frac{66416770678229215641961}{4446374650270322252232153888} a^{13} - \frac{3532961728603864790317}{61755203475976697947668804} a^{12} + \frac{113076294412126051931698883}{4446374650270322252232153888} a^{11} + \frac{342237601627389775911683557}{4446374650270322252232153888} a^{10} + \frac{16850398100573844867730613}{4446374650270322252232153888} a^{9} - \frac{66280841028162562220564641}{494041627807813583581350432} a^{8} + \frac{127464922865202533075049637}{555796831283790281529019236} a^{7} - \frac{13426645987472396425057919}{277898415641895140764509618} a^{6} + \frac{615785415077290512858298817}{1482124883423440750744051296} a^{5} - \frac{666349573188908392325901155}{2223187325135161126116076944} a^{4} + \frac{51155978157667185198882217}{277898415641895140764509618} a^{3} - \frac{39328445759810101738632623}{185265610427930093843006412} a^{2} + \frac{92215175997906627252156059}{277898415641895140764509618} a + \frac{9250046477046870623376583}{46316402606982523460751603}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 34216122.0376 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:S_3$ (as 15T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 150
The 13 conjugacy class representatives for $(C_5^2 : C_3):C_2$
Character table for $(C_5^2 : C_3):C_2$

Intermediate fields

3.1.23.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 15 sibling: data not computed
Degree 25 sibling: data not computed
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{5}$

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
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.10.5.2$x^{10} - 279841 x^{2} + 12872686$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$191$$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
191.2.0.1$x^{2} - x + 19$$1$$2$$0$$C_2$$[\ ]^{2}$
191.2.0.1$x^{2} - x + 19$$1$$2$$0$$C_2$$[\ ]^{2}$
191.10.8.3$x^{10} + 7067 x^{5} + 13169641$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$