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Properties

Label 15.7.13790324594...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{24}\cdot 5^{6}\cdot 19^{4}\cdot 23^{2}\cdot 37^{2}\cdot 43^{2}\cdot 79^{5}\cdot 313^{2}$
Root discriminant $474.21$
Ramified primes $2, 5, 19, 23, 37, 43, 79, 313$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T100

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1216000, 8512000, 14440000, -8010400, -23912640, -3671840, 10176624, 3971312, -712512, -383472, 21512, 7130, -864, 198, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 198*x^13 - 864*x^12 + 7130*x^11 + 21512*x^10 - 383472*x^9 - 712512*x^8 + 3971312*x^7 + 10176624*x^6 - 3671840*x^5 - 23912640*x^4 - 8010400*x^3 + 14440000*x^2 + 8512000*x + 1216000)
 
gp: K = bnfinit(x^15 + 198*x^13 - 864*x^12 + 7130*x^11 + 21512*x^10 - 383472*x^9 - 712512*x^8 + 3971312*x^7 + 10176624*x^6 - 3671840*x^5 - 23912640*x^4 - 8010400*x^3 + 14440000*x^2 + 8512000*x + 1216000, 1)
 

Normalized defining polynomial

\( x^{15} + 198 x^{13} - 864 x^{12} + 7130 x^{11} + 21512 x^{10} - 383472 x^{9} - 712512 x^{8} + 3971312 x^{7} + 10176624 x^{6} - 3671840 x^{5} - 23912640 x^{4} - 8010400 x^{3} + 14440000 x^{2} + 8512000 x + 1216000 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13790324594017072909477729953120256000000=2^{24}\cdot 5^{6}\cdot 19^{4}\cdot 23^{2}\cdot 37^{2}\cdot 43^{2}\cdot 79^{5}\cdot 313^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $474.21$
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:  $2, 5, 19, 23, 37, 43, 79, 313$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{6} - \frac{1}{4} a^{4}$, $\frac{1}{160} a^{11} - \frac{1}{80} a^{9} - \frac{1}{40} a^{8} - \frac{1}{16} a^{7} - \frac{1}{20} a^{6} + \frac{1}{20} a^{5} + \frac{1}{20} a^{4} - \frac{1}{20} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{320} a^{12} + \frac{1}{40} a^{10} - \frac{1}{80} a^{9} + \frac{1}{32} a^{8} + \frac{1}{10} a^{7} - \frac{3}{80} a^{6} - \frac{9}{40} a^{5} - \frac{1}{40} a^{4} + \frac{9}{20} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{3200} a^{13} + \frac{1}{400} a^{11} - \frac{1}{50} a^{10} + \frac{1}{320} a^{9} + \frac{9}{400} a^{8} + \frac{57}{800} a^{7} + \frac{13}{200} a^{6} - \frac{1}{400} a^{5} - \frac{31}{200} a^{4} + \frac{1}{40} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{860446903934238719450390596755318400} a^{14} + \frac{17253232490983696087449972454993}{215111725983559679862597649188829600} a^{13} - \frac{355465952665868476592096011328891}{430223451967119359725195298377659200} a^{12} + \frac{191626835490263048298381391731333}{215111725983559679862597649188829600} a^{11} - \frac{3654748561368852578484287691777759}{430223451967119359725195298377659200} a^{10} - \frac{998320415307872835582728518940461}{107555862991779839931298824594414800} a^{9} + \frac{1985486153218374835058083898888379}{107555862991779839931298824594414800} a^{8} + \frac{12582374012775331442899960292663793}{107555862991779839931298824594414800} a^{7} + \frac{5194291552551125284553259255369193}{53777931495889919965649412297207400} a^{6} - \frac{8342233310118477846286360599144527}{53777931495889919965649412297207400} a^{5} + \frac{3821573952787133285937804390022189}{26888965747944959982824706148603700} a^{4} + \frac{627074029295333537257543547688489}{5377793149588991996564941229720740} a^{3} - \frac{974551497508134262222309384276737}{5377793149588991996564941229720740} a^{2} + \frac{176344249157416320357872899453833}{537779314958899199656494122972074} a + \frac{35575190639032688875033499043898}{268889657479449599828247061486037}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $10$
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:  \( 8395630827600000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T100:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 79 conjugacy class representatives for [1/2.S(5)^3]S(3) are not computed
Character table for [1/2.S(5)^3]S(3) is not computed

Intermediate fields

3.3.316.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.9.1$x^{4} + 6 x^{2} + 2$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.4.9.3$x^{4} + 6 x^{2} + 10$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.6.3.1$x^{6} - 158 x^{4} + 6241 x^{2} - 7888624$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
313Data not computed