Properties

Label 21.3.717...392.1
Degree $21$
Signature $[3, 9]$
Discriminant $-7.175\times 10^{24}$
Root discriminant $15.26$
Ramified primes $2, 7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $F_7$ (as 21T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 42*x^18 + 77*x^17 - 126*x^16 + 168*x^15 - 213*x^14 + 266*x^13 - 280*x^12 + 259*x^11 - 217*x^10 + 133*x^9 - 42*x^8 - 53*x^7 + 126*x^6 - 112*x^5 + 7*x^4 + 63*x^3 - 49*x^2 + 14*x - 1)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 42*x^18 + 77*x^17 - 126*x^16 + 168*x^15 - 213*x^14 + 266*x^13 - 280*x^12 + 259*x^11 - 217*x^10 + 133*x^9 - 42*x^8 - 53*x^7 + 126*x^6 - 112*x^5 + 7*x^4 + 63*x^3 - 49*x^2 + 14*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 14, -49, 63, 7, -112, 126, -53, -42, 133, -217, 259, -280, 266, -213, 168, -126, 77, -42, 21, -7, 1]);
 

\( x^{21} - 7 x^{20} + 21 x^{19} - 42 x^{18} + 77 x^{17} - 126 x^{16} + 168 x^{15} - 213 x^{14} + 266 x^{13} - 280 x^{12} + 259 x^{11} - 217 x^{10} + 133 x^{9} - 42 x^{8} - 53 x^{7} + 126 x^{6} - 112 x^{5} + 7 x^{4} + 63 x^{3} - 49 x^{2} + 14 x - 1 \)

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[3, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-7174552902718171733819392\)\(\medspace = -\,2^{18}\cdot 7^{23}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $15.26$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $3$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{13} a^{19} + \frac{3}{13} a^{18} - \frac{2}{13} a^{17} + \frac{1}{13} a^{15} + \frac{1}{13} a^{14} - \frac{5}{13} a^{13} - \frac{4}{13} a^{12} - \frac{3}{13} a^{11} + \frac{6}{13} a^{10} - \frac{3}{13} a^{9} - \frac{6}{13} a^{8} - \frac{2}{13} a^{7} - \frac{4}{13} a^{6} + \frac{5}{13} a^{3} + \frac{5}{13} a^{2} + \frac{4}{13} a - \frac{1}{13}$, $\frac{1}{17090172314496931} a^{20} + \frac{223953102682691}{17090172314496931} a^{19} + \frac{4978912529283359}{17090172314496931} a^{18} + \frac{4127589721267567}{17090172314496931} a^{17} + \frac{6900728057811901}{17090172314496931} a^{16} - \frac{7419457851442127}{17090172314496931} a^{15} + \frac{7900529489779690}{17090172314496931} a^{14} - \frac{1668847671334546}{17090172314496931} a^{13} - \frac{7902602573222316}{17090172314496931} a^{12} + \frac{267024020850163}{1314628639576687} a^{11} + \frac{4441774245188094}{17090172314496931} a^{10} + \frac{7897737024242235}{17090172314496931} a^{9} + \frac{750035580866610}{17090172314496931} a^{8} - \frac{4496074053875867}{17090172314496931} a^{7} + \frac{6870629632414592}{17090172314496931} a^{6} + \frac{137295108916514}{1314628639576687} a^{5} - \frac{3295518626855801}{17090172314496931} a^{4} + \frac{2550633533191525}{17090172314496931} a^{3} + \frac{3320182089673996}{17090172314496931} a^{2} - \frac{7056387337326788}{17090172314496931} a + \frac{7717614271082649}{17090172314496931}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 10096.3106241 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{9}\cdot 10096.3106241 \cdot 1}{2\sqrt{7174552902718171733819392}}\approx 0.230114554958$ (assuming GRH)

Galois group

$F_7$ (as 21T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.1.52706752.1

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: 7.1.52706752.1
Degree 14 sibling: 14.0.19446011944726528.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
7Data not computed