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Properties

Label 15.7.19978937658...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{24}\cdot 5^{24}\cdot 7^{10}\cdot 29^{4}$
Root discriminant $357.58$
Ramified primes $2, 5, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T98

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![86138816, -82482960, -146316600, 87814465, 18406300, -16746956, 2174580, 693510, -185620, -41050, 7528, 3485, -180, -110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 110*x^13 - 180*x^12 + 3485*x^11 + 7528*x^10 - 41050*x^9 - 185620*x^8 + 693510*x^7 + 2174580*x^6 - 16746956*x^5 + 18406300*x^4 + 87814465*x^3 - 146316600*x^2 - 82482960*x + 86138816)
 
gp: K = bnfinit(x^15 - 110*x^13 - 180*x^12 + 3485*x^11 + 7528*x^10 - 41050*x^9 - 185620*x^8 + 693510*x^7 + 2174580*x^6 - 16746956*x^5 + 18406300*x^4 + 87814465*x^3 - 146316600*x^2 - 82482960*x + 86138816, 1)
 

Normalized defining polynomial

\( x^{15} - 110 x^{13} - 180 x^{12} + 3485 x^{11} + 7528 x^{10} - 41050 x^{9} - 185620 x^{8} + 693510 x^{7} + 2174580 x^{6} - 16746956 x^{5} + 18406300 x^{4} + 87814465 x^{3} - 146316600 x^{2} - 82482960 x + 86138816 \)

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:  \(199789376587969000000000000000000000000=2^{24}\cdot 5^{24}\cdot 7^{10}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $357.58$
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:  $2, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{4} a$, $\frac{1}{325400462845853213181769739085619410520729731746005936} a^{14} + \frac{7299059441925767777517821691748091204199597031103533}{81350115711463303295442434771404852630182432936501484} a^{13} + \frac{779109575200798655840120275540587380329548476124235}{162700231422926606590884869542809705260364865873002968} a^{12} - \frac{7381078579025795663873570683292264787032821882677053}{81350115711463303295442434771404852630182432936501484} a^{11} + \frac{25277518743392829563010694762197896999467508261662709}{325400462845853213181769739085619410520729731746005936} a^{10} - \frac{1894824972458318890967858677930754599190497128018938}{20337528927865825823860608692851213157545608234125371} a^{9} + \frac{39746041221979724017633341415484898412294934148428743}{162700231422926606590884869542809705260364865873002968} a^{8} + \frac{4593310352177370991147520784265039147435228664069397}{40675057855731651647721217385702426315091216468250742} a^{7} - \frac{3195346625282521242884771733634730442818839636371497}{162700231422926606590884869542809705260364865873002968} a^{6} - \frac{32517299845380554061941207238724168839643801644947943}{81350115711463303295442434771404852630182432936501484} a^{5} + \frac{2697547285570301086887714347656579133058146592314410}{20337528927865825823860608692851213157545608234125371} a^{4} - \frac{13945250775033409608025906191257486400898501245606199}{40675057855731651647721217385702426315091216468250742} a^{3} + \frac{30337167096489383707221449055222987731487730679394893}{325400462845853213181769739085619410520729731746005936} a^{2} - \frac{21519973778793684815422439551593293183069764440327311}{81350115711463303295442434771404852630182432936501484} a - \frac{9259249556192883439696670648508977236213897265320294}{20337528927865825823860608692851213157545608234125371}$

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:  \( 41444248102300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T98:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 71 conjugacy class representatives for [1/2.S(5)^3]3 are not computed
Character table for [1/2.S(5)^3]3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $15$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $15$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.24.227$x^{12} + 4 x^{11} + 2 x^{10} - 4 x^{9} - 8 x^{8} - 8 x^{7} - 4 x^{6} + 12 x^{4} + 16 x^{3} - 8 x^{2} - 8$$4$$3$$24$12T89$[2, 2, 2, 3, 3]^{6}$
5Data not computed
7Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$