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Properties

Label 15.15.1683683687...0625.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{24}\cdot 7^{10}$
Root discriminant $48.06$
Ramified primes $5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{15}$ (as 15T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![301, -770, -3500, 3765, 11735, -4757, -13645, 3115, 6825, -1350, -1539, 305, 150, -30, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 30*x^13 + 150*x^12 + 305*x^11 - 1539*x^10 - 1350*x^9 + 6825*x^8 + 3115*x^7 - 13645*x^6 - 4757*x^5 + 11735*x^4 + 3765*x^3 - 3500*x^2 - 770*x + 301)
 
gp: K = bnfinit(x^15 - 5*x^14 - 30*x^13 + 150*x^12 + 305*x^11 - 1539*x^10 - 1350*x^9 + 6825*x^8 + 3115*x^7 - 13645*x^6 - 4757*x^5 + 11735*x^4 + 3765*x^3 - 3500*x^2 - 770*x + 301, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 30 x^{13} + 150 x^{12} + 305 x^{11} - 1539 x^{10} - 1350 x^{9} + 6825 x^{8} + 3115 x^{7} - 13645 x^{6} - 4757 x^{5} + 11735 x^{4} + 3765 x^{3} - 3500 x^{2} - 770 x + 301 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(16836836874485015869140625=5^{24}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.06$
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:  $5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(175=5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{175}(1,·)$, $\chi_{175}(36,·)$, $\chi_{175}(71,·)$, $\chi_{175}(106,·)$, $\chi_{175}(11,·)$, $\chi_{175}(141,·)$, $\chi_{175}(46,·)$, $\chi_{175}(16,·)$, $\chi_{175}(81,·)$, $\chi_{175}(51,·)$, $\chi_{175}(116,·)$, $\chi_{175}(86,·)$, $\chi_{175}(151,·)$, $\chi_{175}(121,·)$, $\chi_{175}(156,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{8} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{49} a^{13} - \frac{1}{49} a^{12} - \frac{3}{49} a^{11} + \frac{16}{49} a^{10} - \frac{11}{49} a^{9} - \frac{16}{49} a^{8} + \frac{9}{49} a^{7} + \frac{16}{49} a^{6} + \frac{1}{7} a^{5} - \frac{17}{49} a^{4} + \frac{5}{49} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{7659219376252355849} a^{14} - \frac{40518519814687833}{7659219376252355849} a^{13} + \frac{74267797811977264}{7659219376252355849} a^{12} - \frac{3084242977174696220}{7659219376252355849} a^{11} + \frac{1731699461148162129}{7659219376252355849} a^{10} + \frac{2912998882946688307}{7659219376252355849} a^{9} - \frac{2154574297594152885}{7659219376252355849} a^{8} - \frac{28034412114270276}{156310599515354201} a^{7} - \frac{3083603310104332976}{7659219376252355849} a^{6} - \frac{3533884455897250815}{7659219376252355849} a^{5} + \frac{2849789317377109395}{7659219376252355849} a^{4} - \frac{763865364125619810}{7659219376252355849} a^{3} - \frac{254537904435646276}{1094174196607479407} a^{2} - \frac{124348582994085482}{1094174196607479407} a - \frac{8846912733002633}{25445911549011149}$

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:  $14$
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:  \( 87241496.8826 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{15}$ (as 15T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 15
The 15 conjugacy class representatives for $C_{15}$
Character table for $C_{15}$

Intermediate fields

\(\Q(\zeta_{7})^+\), 5.5.390625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $15$ $15$ R R $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ $15$ $15$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{15}$ $15$ $15$ $15$

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
5Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$