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Properties

Label 16.0.48646548869...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 109^{8}$
Root discriminant $34.91$
Ramified primes $5, 109$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -297, 765, -1671, 3238, -5961, 9764, -13559, 13555, 12, -1287, 154, 57, 29, -8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 29*x^13 + 57*x^12 + 154*x^11 - 1287*x^10 + 12*x^9 + 13555*x^8 - 13559*x^7 + 9764*x^6 - 5961*x^5 + 3238*x^4 - 1671*x^3 + 765*x^2 - 297*x + 81)
 
gp: K = bnfinit(x^16 - 2*x^15 - 8*x^14 + 29*x^13 + 57*x^12 + 154*x^11 - 1287*x^10 + 12*x^9 + 13555*x^8 - 13559*x^7 + 9764*x^6 - 5961*x^5 + 3238*x^4 - 1671*x^3 + 765*x^2 - 297*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 8 x^{14} + 29 x^{13} + 57 x^{12} + 154 x^{11} - 1287 x^{10} + 12 x^{9} + 13555 x^{8} - 13559 x^{7} + 9764 x^{6} - 5961 x^{5} + 3238 x^{4} - 1671 x^{3} + 765 x^{2} - 297 x + 81 \)

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:  \(4864654886938945556640625=5^{12}\cdot 109^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.91$
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:  $5, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{11} + \frac{1}{10} a^{10} + \frac{2}{5} a^{8} - \frac{3}{10} a^{7} - \frac{3}{10} a^{6} + \frac{3}{10} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a + \frac{1}{10}$, $\frac{1}{5517177630} a^{13} + \frac{16772447}{1103435526} a^{12} - \frac{233892601}{1103435526} a^{11} + \frac{1332579179}{5517177630} a^{10} - \frac{98073691}{919529605} a^{9} - \frac{1499038637}{5517177630} a^{8} - \frac{76418669}{367811842} a^{7} - \frac{314155303}{1839059210} a^{6} + \frac{791977903}{5517177630} a^{5} - \frac{377028376}{2758588815} a^{4} + \frac{20985377}{5517177630} a^{3} - \frac{285517117}{1839059210} a^{2} - \frac{1907598239}{5517177630} a - \frac{142129697}{1839059210}$, $\frac{1}{16551532890} a^{14} + \frac{1}{16551532890} a^{13} - \frac{334798108}{8275766445} a^{12} + \frac{1638325249}{8275766445} a^{11} + \frac{82325199}{919529605} a^{10} - \frac{3870956933}{16551532890} a^{9} + \frac{2562488443}{5517177630} a^{8} + \frac{986119619}{2758588815} a^{7} - \frac{3368864569}{8275766445} a^{6} + \frac{968930504}{8275766445} a^{5} - \frac{3165677521}{16551532890} a^{4} + \frac{1840760749}{5517177630} a^{3} + \frac{1513437452}{8275766445} a^{2} + \frac{138171770}{551717763} a + \frac{191573341}{919529605}$, $\frac{1}{248272993350} a^{15} - \frac{1}{49654598670} a^{14} + \frac{7}{248272993350} a^{13} - \frac{5362978811}{124136496675} a^{12} - \frac{3211813159}{82757664450} a^{11} + \frac{4357961147}{49654598670} a^{10} - \frac{17357864479}{82757664450} a^{9} - \frac{12555795449}{82757664450} a^{8} - \frac{19278697522}{124136496675} a^{7} + \frac{108998743573}{248272993350} a^{6} + \frac{7411811383}{49654598670} a^{5} + \frac{13790187131}{27585888150} a^{4} + \frac{33697179061}{248272993350} a^{3} - \frac{285099742}{725944425} a^{2} + \frac{12624735343}{27585888150} a + \frac{2147377828}{4597648025}$

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:  \( \frac{530854489}{49654598670} a^{15} - \frac{48259499}{2613399930} a^{14} - \frac{2267455384}{24827299335} a^{13} + \frac{2847310441}{9930919734} a^{12} + \frac{5742880381}{8275766445} a^{11} + \frac{18000793127}{9930919734} a^{10} - \frac{44060922587}{3310306578} a^{9} - \frac{29865124069}{8275766445} a^{8} + \frac{7197469940359}{49654598670} a^{7} - \frac{2617691744758}{24827299335} a^{6} + \frac{3220211589773}{49654598670} a^{5} - \frac{194534040469}{5517177630} a^{4} + \frac{106383282829}{4965459867} a^{3} - \frac{15491299179}{1839059210} a^{2} + \frac{3040348437}{919529605} a - \frac{868670982}{919529605} \) (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:  \( 589962.086805 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{545}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{109}) \), \(\Q(\sqrt{5}, \sqrt{109})\), 4.0.2725.1 x2, 4.0.59405.1 x2, 4.4.1485125.1 x2, 4.4.13625.1 x2, 4.0.1485125.1, \(\Q(\zeta_{5})\), 8.0.88223850625.1, 8.8.2205596265625.1, 8.0.2205596265625.3, 8.0.2205596265625.2 x2, 8.0.185640625.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$109$109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$