Properties

Label 16.0.23612624896...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $25.02$
Ramified primes $2, 5, 7$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -64, 184, -464, 1084, -1816, 2758, -3712, 3711, 10, -497, 88, 19, 20, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 5*x^14 + 20*x^13 + 19*x^12 + 88*x^11 - 497*x^10 + 10*x^9 + 3711*x^8 - 3712*x^7 + 2758*x^6 - 1816*x^5 + 1084*x^4 - 464*x^3 + 184*x^2 - 64*x + 16)
 
gp: K = bnfinit(x^16 - 2*x^15 - 5*x^14 + 20*x^13 + 19*x^12 + 88*x^11 - 497*x^10 + 10*x^9 + 3711*x^8 - 3712*x^7 + 2758*x^6 - 1816*x^5 + 1084*x^4 - 464*x^3 + 184*x^2 - 64*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 5 x^{14} + 20 x^{13} + 19 x^{12} + 88 x^{11} - 497 x^{10} + 10 x^{9} + 3711 x^{8} - 3712 x^{7} + 2758 x^{6} - 1816 x^{5} + 1084 x^{4} - 464 x^{3} + 184 x^{2} - 64 x + 16 \)

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:  \(23612624896000000000000=2^{24}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.02$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(280=2^{3}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{280}(1,·)$, $\chi_{280}(267,·)$, $\chi_{280}(139,·)$, $\chi_{280}(209,·)$, $\chi_{280}(83,·)$, $\chi_{280}(153,·)$, $\chi_{280}(27,·)$, $\chi_{280}(97,·)$, $\chi_{280}(99,·)$, $\chi_{280}(41,·)$, $\chi_{280}(43,·)$, $\chi_{280}(113,·)$, $\chi_{280}(211,·)$, $\chi_{280}(169,·)$, $\chi_{280}(57,·)$, $\chi_{280}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11} a^{9} - \frac{4}{11} a^{8} + \frac{5}{11} a^{7} + \frac{2}{11} a^{6} + \frac{3}{11} a^{5} - \frac{1}{11} a^{4} + \frac{4}{11} a^{3} - \frac{5}{11} a^{2} - \frac{2}{11} a - \frac{3}{11}$, $\frac{1}{22} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{5}{11}$, $\frac{1}{22} a^{11} - \frac{1}{22} a^{9} + \frac{2}{11} a^{8} - \frac{5}{22} a^{7} - \frac{1}{11} a^{6} - \frac{3}{22} a^{5} - \frac{5}{11} a^{4} + \frac{7}{22} a^{3} - \frac{3}{11} a^{2} - \frac{5}{11} a - \frac{4}{11}$, $\frac{1}{44} a^{12} - \frac{1}{44} a^{10} - \frac{1}{22} a^{9} + \frac{19}{44} a^{8} - \frac{5}{22} a^{7} + \frac{7}{44} a^{6} + \frac{4}{11} a^{5} - \frac{9}{44} a^{4} + \frac{7}{22} a^{3} - \frac{1}{22} a^{2} + \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{1254836} a^{13} + \frac{10955}{1254836} a^{12} - \frac{3445}{1254836} a^{11} + \frac{107}{6004} a^{10} + \frac{12713}{1254836} a^{9} - \frac{364297}{1254836} a^{8} - \frac{418763}{1254836} a^{7} - \frac{276315}{1254836} a^{6} - \frac{318569}{1254836} a^{5} - \frac{326345}{1254836} a^{4} + \frac{4336}{16511} a^{3} + \frac{251633}{627418} a^{2} + \frac{109034}{313709} a + \frac{7419}{313709}$, $\frac{1}{27606392} a^{14} - \frac{1}{6901598} a^{13} - \frac{13209}{1452968} a^{12} + \frac{236633}{13803196} a^{11} + \frac{57401}{27606392} a^{10} + \frac{627417}{13803196} a^{9} + \frac{11543749}{27606392} a^{8} + \frac{188411}{6901598} a^{7} + \frac{12791}{349448} a^{6} + \frac{3381067}{13803196} a^{5} - \frac{1003923}{6901598} a^{4} + \frac{1945019}{6901598} a^{3} + \frac{564667}{6901598} a^{2} + \frac{9569}{181621} a + \frac{828475}{3450799}$, $\frac{1}{27606392} a^{15} - \frac{1}{2509672} a^{13} + \frac{4391}{627418} a^{12} - \frac{53593}{2509672} a^{11} + \frac{68985}{3450799} a^{10} - \frac{12713}{2509672} a^{9} + \frac{339003}{1254836} a^{8} + \frac{418763}{2509672} a^{7} - \frac{166203}{627418} a^{6} - \frac{71648}{181621} a^{5} - \frac{307391}{1254836} a^{4} + \frac{12175}{33022} a^{3} - \frac{54519}{627418} a^{2} + \frac{31040}{313709} a + \frac{1338806}{3450799}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( \frac{13159}{13803196} a^{15} + \frac{65795}{27606392} a^{14} - \frac{65795}{6901598} a^{13} - \frac{13159}{1452968} a^{12} + \frac{1145401}{13803196} a^{11} + \frac{6540023}{27606392} a^{10} - \frac{65795}{13803196} a^{9} - \frac{48833049}{27606392} a^{8} + \frac{6105776}{3450799} a^{7} + \frac{4179365}{349448} a^{6} + \frac{2987093}{3450799} a^{5} - \frac{3566089}{6901598} a^{4} + \frac{763222}{3450799} a^{3} - \frac{302657}{3450799} a^{2} + \frac{407678}{181621} a - \frac{26318}{3450799} \) (order $10$)
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:  \( 85253.3675979 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-10}, \sqrt{14})\), \(\Q(\sqrt{5}, \sqrt{14})\), \(\Q(\sqrt{-7}, \sqrt{-10})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), 4.0.392000.2, 4.4.8000.1, \(\Q(\zeta_{5})\), 4.4.6125.1, 8.0.6146560000.1, 8.0.153664000000.4, 8.0.37515625.1, 8.0.153664000000.6, 8.8.153664000000.2, 8.0.153664000000.3, 8.0.64000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{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}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$