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Properties

Label 16.0.27204384060...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 19^{8}$
Root discriminant $25.24$
Ramified primes $3, 5, 19$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![390625, -78125, 78125, -43750, 8750, 3875, -750, 2170, -1059, 434, -30, 31, 14, -14, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 5*x^14 - 14*x^13 + 14*x^12 + 31*x^11 - 30*x^10 + 434*x^9 - 1059*x^8 + 2170*x^7 - 750*x^6 + 3875*x^5 + 8750*x^4 - 43750*x^3 + 78125*x^2 - 78125*x + 390625)
 
gp: K = bnfinit(x^16 - x^15 + 5*x^14 - 14*x^13 + 14*x^12 + 31*x^11 - 30*x^10 + 434*x^9 - 1059*x^8 + 2170*x^7 - 750*x^6 + 3875*x^5 + 8750*x^4 - 43750*x^3 + 78125*x^2 - 78125*x + 390625, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 5 x^{14} - 14 x^{13} + 14 x^{12} + 31 x^{11} - 30 x^{10} + 434 x^{9} - 1059 x^{8} + 2170 x^{7} - 750 x^{6} + 3875 x^{5} + 8750 x^{4} - 43750 x^{3} + 78125 x^{2} - 78125 x + 390625 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(27204384060547119140625=3^{8}\cdot 5^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.24$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(285=3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{285}(1,·)$, $\chi_{285}(134,·)$, $\chi_{285}(77,·)$, $\chi_{285}(208,·)$, $\chi_{285}(248,·)$, $\chi_{285}(151,·)$, $\chi_{285}(284,·)$, $\chi_{285}(94,·)$, $\chi_{285}(37,·)$, $\chi_{285}(227,·)$, $\chi_{285}(229,·)$, $\chi_{285}(172,·)$, $\chi_{285}(113,·)$, $\chi_{285}(56,·)$, $\chi_{285}(58,·)$, $\chi_{285}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{275} a^{10} - \frac{1}{25} a^{9} - \frac{2}{5} a^{8} + \frac{1}{25} a^{7} - \frac{11}{25} a^{6} - \frac{109}{275} a^{5} + \frac{2}{5} a^{4} - \frac{6}{25} a^{3} - \frac{9}{25} a^{2} + \frac{2}{5} a + \frac{4}{11}$, $\frac{1}{1375} a^{11} - \frac{1}{1375} a^{10} + \frac{1}{25} a^{9} + \frac{51}{125} a^{8} + \frac{24}{125} a^{7} + \frac{331}{1375} a^{6} + \frac{134}{275} a^{5} + \frac{44}{125} a^{4} + \frac{6}{125} a^{3} + \frac{9}{25} a^{2} - \frac{18}{55} a - \frac{3}{11}$, $\frac{1}{27500} a^{12} + \frac{1}{6875} a^{11} + \frac{1}{2500} a^{9} + \frac{226}{625} a^{8} - \frac{756}{6875} a^{7} + \frac{1}{220} a^{6} - \frac{164}{625} a^{5} - \frac{131}{625} a^{4} + \frac{1}{4} a^{3} + \frac{1}{275} a^{2} + \frac{14}{55} a - \frac{1}{4}$, $\frac{1}{243237500} a^{13} + \frac{629}{243237500} a^{12} - \frac{72}{486475} a^{11} + \frac{3501}{22112500} a^{10} - \frac{820971}{22112500} a^{9} + \frac{14323994}{60809375} a^{8} + \frac{327069}{1945900} a^{7} + \frac{48514909}{243237500} a^{6} + \frac{1234244}{5528125} a^{5} - \frac{33807}{176900} a^{4} - \frac{2004361}{9729500} a^{3} + \frac{217994}{486475} a^{2} + \frac{134413}{389180} a + \frac{1803}{7076}$, $\frac{1}{1216187500} a^{14} - \frac{1}{1216187500} a^{13} + \frac{499}{60809375} a^{12} - \frac{278489}{1216187500} a^{11} + \frac{318389}{1216187500} a^{10} - \frac{21552561}{304046875} a^{9} + \frac{84062849}{243237500} a^{8} + \frac{323550659}{1216187500} a^{7} - \frac{18720796}{304046875} a^{6} + \frac{41316639}{243237500} a^{5} + \frac{13932769}{48647500} a^{4} + \frac{784951}{2432375} a^{3} - \frac{304223}{1945900} a^{2} - \frac{61173}{389180} a - \frac{4705}{19459}$, $\frac{1}{6080937500} a^{15} - \frac{1}{6080937500} a^{14} + \frac{1}{1216187500} a^{13} + \frac{40493}{3040468750} a^{12} + \frac{1195889}{6080937500} a^{11} + \frac{2850031}{6080937500} a^{10} + \frac{33321597}{608093750} a^{9} + \frac{1328617559}{6080937500} a^{8} + \frac{2921404941}{6080937500} a^{7} + \frac{173040217}{608093750} a^{6} + \frac{15754301}{48647500} a^{5} + \frac{224859}{48647500} a^{4} + \frac{633277}{4864750} a^{3} - \frac{930101}{1945900} a^{2} + \frac{35505}{77836} a + \frac{1139}{77836}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( \frac{1}{389180} a^{15} - \frac{1}{77836} a^{14} + \frac{7}{194590} a^{13} - \frac{7}{194590} a^{12} - \frac{31}{389180} a^{11} + \frac{3}{38918} a^{10} - \frac{217}{194590} a^{9} + \frac{1059}{389180} a^{8} - \frac{217}{38918} a^{7} + \frac{75}{38918} a^{6} - \frac{775}{77836} a^{5} - \frac{875}{38918} a^{4} + \frac{4375}{38918} a^{3} - \frac{15625}{77836} a^{2} - \frac{1253}{194590} a - \frac{78125}{77836} \) (order $30$)
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:  \( 49344.1781525 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{285}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{-95})\), \(\Q(\sqrt{5}, \sqrt{57})\), \(\Q(\sqrt{-15}, \sqrt{-19})\), \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\sqrt{-15}, \sqrt{57})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-19})\), 4.0.406125.2, \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), 4.4.45125.1, 8.0.6597500625.1, 8.0.164937515625.3, 8.8.164937515625.1, 8.0.164937515625.1, 8.0.2036265625.1, 8.0.164937515625.2, \(\Q(\zeta_{15})\)

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$