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Properties

Label 20.0.35309406125...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{23}\cdot 7^{10}$
Root discriminant $33.68$
Ramified primes $2, 5, 7$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3920, 0, -8400, 0, 5420, 0, -2980, 0, 1745, 0, 815, 0, -385, 0, 30, 0, 75, 0, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 15*x^18 + 75*x^16 + 30*x^14 - 385*x^12 + 815*x^10 + 1745*x^8 - 2980*x^6 + 5420*x^4 - 8400*x^2 + 3920)
 
gp: K = bnfinit(x^20 + 15*x^18 + 75*x^16 + 30*x^14 - 385*x^12 + 815*x^10 + 1745*x^8 - 2980*x^6 + 5420*x^4 - 8400*x^2 + 3920, 1)
 

Normalized defining polynomial

\( x^{20} + 15 x^{18} + 75 x^{16} + 30 x^{14} - 385 x^{12} + 815 x^{10} + 1745 x^{8} - 2980 x^{6} + 5420 x^{4} - 8400 x^{2} + 3920 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3530940612500000000000000000000=2^{20}\cdot 5^{23}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.68$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{8} a^{9} - \frac{3}{16} a^{7} - \frac{5}{16} a^{5} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{304} a^{14} - \frac{13}{304} a^{12} + \frac{11}{152} a^{10} - \frac{59}{304} a^{8} + \frac{7}{304} a^{6} - \frac{1}{2} a^{5} + \frac{7}{152} a^{4} - \frac{1}{2} a^{3} + \frac{13}{76} a^{2} - \frac{1}{2} a + \frac{5}{19}$, $\frac{1}{608} a^{15} - \frac{13}{608} a^{13} + \frac{11}{304} a^{11} + \frac{93}{608} a^{9} - \frac{1}{4} a^{8} - \frac{145}{608} a^{7} - \frac{1}{2} a^{6} + \frac{83}{304} a^{5} - \frac{1}{2} a^{4} + \frac{13}{152} a^{3} + \frac{1}{4} a^{2} - \frac{7}{19} a$, $\frac{1}{6688} a^{16} - \frac{9}{6688} a^{14} - \frac{53}{3344} a^{12} + \frac{713}{6688} a^{10} + \frac{145}{608} a^{8} - \frac{1005}{3344} a^{6} - \frac{1}{2} a^{5} + \frac{97}{418} a^{4} - \frac{1}{2} a^{3} + \frac{93}{836} a^{2} - \frac{1}{2} a - \frac{151}{418}$, $\frac{1}{13376} a^{17} - \frac{1}{13376} a^{16} - \frac{9}{13376} a^{15} - \frac{13}{13376} a^{14} - \frac{53}{6688} a^{13} + \frac{49}{1672} a^{12} + \frac{713}{13376} a^{11} - \frac{63}{704} a^{10} - \frac{159}{1216} a^{9} + \frac{277}{1216} a^{8} + \frac{667}{6688} a^{7} + \frac{29}{209} a^{6} - \frac{28}{209} a^{5} + \frac{1207}{3344} a^{4} + \frac{93}{1672} a^{3} - \frac{327}{836} a^{2} + \frac{267}{836} a - \frac{377}{836}$, $\frac{1}{94176750976} a^{18} + \frac{143229}{23544187744} a^{16} + \frac{154480695}{94176750976} a^{14} + \frac{3260419857}{94176750976} a^{12} - \frac{326743795}{3363455392} a^{10} - \frac{1}{4} a^{9} - \frac{13894529261}{94176750976} a^{8} + \frac{1381282631}{5886046936} a^{6} + \frac{10475981177}{23544187744} a^{4} - \frac{1}{4} a^{3} + \frac{60293853}{133773794} a^{2} + \frac{252328045}{840863848}$, $\frac{1}{188353501952} a^{19} - \frac{1}{188353501952} a^{18} + \frac{143229}{47088375488} a^{17} - \frac{143229}{47088375488} a^{16} + \frac{154480695}{188353501952} a^{15} + \frac{155311249}{188353501952} a^{14} + \frac{3260419857}{188353501952} a^{13} + \frac{4484378743}{188353501952} a^{12} - \frac{326743795}{6726910784} a^{11} + \frac{149719827}{6726910784} a^{10} - \frac{13894529261}{188353501952} a^{9} + \frac{19160992309}{188353501952} a^{8} - \frac{1561740837}{11772093872} a^{7} + \frac{4866061359}{23544187744} a^{6} + \frac{22248075049}{47088375488} a^{5} - \frac{562638969}{47088375488} a^{4} - \frac{1648261}{66886897} a^{3} + \frac{129805749}{267547588} a^{2} + \frac{252328045}{1681727696} a - \frac{451480009}{1681727696}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 6938252.87449 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.98000.1, 5.1.2450000.1 x5, 10.2.30012500000000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.2450000.1
Degree 10 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
5Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$