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Properties

Label 15.7.15630956656...1536.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{26}\cdot 137^{10}$
Root discriminant $88.36$
Ramified primes $2, 137$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_8$ (as 15T72)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7288, -36216, -59864, -18504, 42844, 28484, -3724, 1548, 3878, -1650, -906, 394, 81, -37, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 37*x^13 + 81*x^12 + 394*x^11 - 906*x^10 - 1650*x^9 + 3878*x^8 + 1548*x^7 - 3724*x^6 + 28484*x^5 + 42844*x^4 - 18504*x^3 - 59864*x^2 - 36216*x - 7288)
 
gp: K = bnfinit(x^15 - x^14 - 37*x^13 + 81*x^12 + 394*x^11 - 906*x^10 - 1650*x^9 + 3878*x^8 + 1548*x^7 - 3724*x^6 + 28484*x^5 + 42844*x^4 - 18504*x^3 - 59864*x^2 - 36216*x - 7288, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 37 x^{13} + 81 x^{12} + 394 x^{11} - 906 x^{10} - 1650 x^{9} + 3878 x^{8} + 1548 x^{7} - 3724 x^{6} + 28484 x^{5} + 42844 x^{4} - 18504 x^{3} - 59864 x^{2} - 36216 x - 7288 \)

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:  \(156309566567541201405567041536=2^{26}\cdot 137^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.36$
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:  $2, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{20642281406186352666640} a^{14} + \frac{190615496248685739383}{20642281406186352666640} a^{13} - \frac{83273540404475509397}{4128456281237270533328} a^{12} - \frac{281413463380118206219}{20642281406186352666640} a^{11} + \frac{591209531601079249547}{5160570351546588166660} a^{10} + \frac{166110502751627877159}{5160570351546588166660} a^{9} + \frac{641510565995572699131}{5160570351546588166660} a^{8} - \frac{129576052009406554249}{5160570351546588166660} a^{7} - \frac{147334171999833970566}{1290142587886647041665} a^{6} + \frac{1205966499467179794853}{5160570351546588166660} a^{5} - \frac{718941337072490358537}{5160570351546588166660} a^{4} - \frac{2545315643996386761567}{5160570351546588166660} a^{3} - \frac{1257009904826650748957}{2580285175773294083330} a^{2} + \frac{321279110134744640497}{1290142587886647041665} a + \frac{323797048716766634962}{1290142587886647041665}$

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:  \( 12284484047.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_8$ (as 15T72):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 20160
The 14 conjugacy class representatives for $A_8$
Character table for $A_8$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 8 sibling: data not computed
Degree 28 sibling: data not computed
Degree 35 sibling: data not computed
Arithmetically equvalently 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 $15$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.26.106$x^{12} + 4 x^{8} + 4 x^{6} + 4 x^{5} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 2$$12$$1$$26$12T66$[4/3, 4/3, 8/3, 8/3]_{3}^{2}$
$137$137.6.4.2$x^{6} - 137 x^{3} + 112614$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
137.9.6.1$x^{9} - 18769 x^{3} + 12856765$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$