Properties

Label 25.5.953...000.1
Degree $25$
Signature $[5, 10]$
Discriminant $9.537\times 10^{33}$
Root discriminant $22.87$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_5\times D_5$ (as 25T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 + 10*x^23 - 10*x^22 + 17*x^20 - 25*x^19 + 10*x^18 + 55*x^17 - 150*x^16 + 156*x^15 - 100*x^14 + 115*x^13 - 60*x^12 - 175*x^11 + 208*x^10 + 70*x^9 - 80*x^8 - 70*x^7 - 25*x^6 + 20*x^5 + 15*x^4 + 15*x^3 + 10*x^2 - 1)
 
gp: K = bnfinit(x^25 - 5*x^24 + 10*x^23 - 10*x^22 + 17*x^20 - 25*x^19 + 10*x^18 + 55*x^17 - 150*x^16 + 156*x^15 - 100*x^14 + 115*x^13 - 60*x^12 - 175*x^11 + 208*x^10 + 70*x^9 - 80*x^8 - 70*x^7 - 25*x^6 + 20*x^5 + 15*x^4 + 15*x^3 + 10*x^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 10, 15, 15, 20, -25, -70, -80, 70, 208, -175, -60, 115, -100, 156, -150, 55, 10, -25, 17, 0, -10, 10, -5, 1]);
 

\(x^{25} - 5 x^{24} + 10 x^{23} - 10 x^{22} + 17 x^{20} - 25 x^{19} + 10 x^{18} + 55 x^{17} - 150 x^{16} + 156 x^{15} - 100 x^{14} + 115 x^{13} - 60 x^{12} - 175 x^{11} + 208 x^{10} + 70 x^{9} - 80 x^{8} - 70 x^{7} - 25 x^{6} + 20 x^{5} + 15 x^{4} + 15 x^{3} + 10 x^{2} - 1\)  Toggle raw display

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

Invariants

Degree:  $25$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[5, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(9536743164062500000000000000000000\)\(\medspace = 2^{20}\cdot 5^{40}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $22.87$
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:  $2, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $5$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7} a^{18} - \frac{3}{7} a^{17} + \frac{2}{7} a^{16} + \frac{1}{7} a^{15} - \frac{1}{7} a^{14} + \frac{2}{7} a^{13} - \frac{2}{7} a^{12} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{19} + \frac{2}{7} a^{15} - \frac{1}{7} a^{14} - \frac{3}{7} a^{13} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{20} + \frac{2}{7} a^{16} - \frac{1}{7} a^{15} - \frac{3}{7} a^{14} - \frac{1}{7} a^{11} - \frac{3}{7} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{7} a^{21} + \frac{2}{7} a^{17} - \frac{1}{7} a^{16} - \frac{3}{7} a^{15} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2}$, $\frac{1}{301} a^{22} + \frac{16}{301} a^{21} + \frac{3}{301} a^{20} + \frac{17}{301} a^{19} + \frac{1}{301} a^{18} - \frac{15}{301} a^{17} - \frac{148}{301} a^{16} + \frac{17}{301} a^{15} - \frac{74}{301} a^{14} - \frac{54}{301} a^{13} - \frac{87}{301} a^{12} + \frac{61}{301} a^{11} + \frac{118}{301} a^{10} - \frac{6}{301} a^{9} + \frac{6}{43} a^{8} + \frac{108}{301} a^{7} + \frac{69}{301} a^{6} - \frac{148}{301} a^{5} + \frac{90}{301} a^{4} + \frac{3}{43} a^{3} - \frac{52}{301} a^{2} - \frac{17}{301} a - \frac{110}{301}$, $\frac{1}{18737551} a^{23} + \frac{8378}{18737551} a^{22} + \frac{1093985}{18737551} a^{21} - \frac{613619}{18737551} a^{20} + \frac{305168}{18737551} a^{19} + \frac{516865}{18737551} a^{18} + \frac{1077043}{2676793} a^{17} - \frac{8950426}{18737551} a^{16} + \frac{9172553}{18737551} a^{15} - \frac{809215}{2676793} a^{14} - \frac{608714}{18737551} a^{13} + \frac{4974453}{18737551} a^{12} - \frac{2701599}{18737551} a^{11} - \frac{4277866}{18737551} a^{10} + \frac{3918856}{18737551} a^{9} + \frac{3623784}{18737551} a^{8} + \frac{815574}{18737551} a^{7} + \frac{668869}{2676793} a^{6} - \frac{2115030}{18737551} a^{5} - \frac{5588738}{18737551} a^{4} - \frac{4395135}{18737551} a^{3} - \frac{2394695}{18737551} a^{2} - \frac{6760265}{18737551} a - \frac{3989332}{18737551}$, $\frac{1}{5640002851} a^{24} + \frac{120}{5640002851} a^{23} + \frac{3310358}{5640002851} a^{22} - \frac{383907032}{5640002851} a^{21} - \frac{232417019}{5640002851} a^{20} + \frac{396712118}{5640002851} a^{19} + \frac{262176787}{5640002851} a^{18} + \frac{1112295315}{5640002851} a^{17} - \frac{1686406609}{5640002851} a^{16} + \frac{1019061081}{5640002851} a^{15} + \frac{60076050}{131162857} a^{14} - \frac{1567680598}{5640002851} a^{13} - \frac{1627610932}{5640002851} a^{12} - \frac{722549997}{5640002851} a^{11} + \frac{2289759767}{5640002851} a^{10} + \frac{1379885406}{5640002851} a^{9} + \frac{2104307061}{5640002851} a^{8} + \frac{1519112260}{5640002851} a^{7} - \frac{42682088}{131162857} a^{6} - \frac{2267137400}{5640002851} a^{5} + \frac{1390615046}{5640002851} a^{4} - \frac{1647798090}{5640002851} a^{3} + \frac{16388745}{5640002851} a^{2} + \frac{2108505680}{5640002851} a - \frac{650346502}{5640002851}$  Toggle raw display

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:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
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:  \( 22714715.777837068 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{5}\cdot(2\pi)^{10}\cdot 22714715.777837068 \cdot 1}{2\sqrt{9536743164062500000000000000000000}}\approx 0.356883056611535$ (assuming GRH)

Galois group

$C_5\times D_5$ (as 25T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 50
The 20 conjugacy class representatives for $C_5\times D_5$
Character table for $C_5\times D_5$

Intermediate fields

5.5.390625.1, 5.1.6250000.1

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

Sibling fields

Degree 10 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
5Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.25.5t1.a.d$1$ $ 5^{2}$ 5.5.390625.1 $C_5$ (as 5T1) $0$ $1$
1.100.10t1.a.b$1$ $ 2^{2} \cdot 5^{2}$ 10.0.156250000000000.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.25.5t1.a.a$1$ $ 5^{2}$ 5.5.390625.1 $C_5$ (as 5T1) $0$ $1$
* 1.25.5t1.a.b$1$ $ 5^{2}$ 5.5.390625.1 $C_5$ (as 5T1) $0$ $1$
1.100.10t1.a.c$1$ $ 2^{2} \cdot 5^{2}$ 10.0.156250000000000.1 $C_{10}$ (as 10T1) $0$ $-1$
1.100.10t1.a.a$1$ $ 2^{2} \cdot 5^{2}$ 10.0.156250000000000.1 $C_{10}$ (as 10T1) $0$ $-1$
1.100.10t1.a.d$1$ $ 2^{2} \cdot 5^{2}$ 10.0.156250000000000.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.25.5t1.a.c$1$ $ 5^{2}$ 5.5.390625.1 $C_5$ (as 5T1) $0$ $1$
* 2.2500.5t2.a.a$2$ $ 2^{2} \cdot 5^{4}$ 5.1.6250000.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2500.10t6.a.d$2$ $ 2^{2} \cdot 5^{4}$ 25.5.9536743164062500000000000000000000.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.2500.10t6.a.a$2$ $ 2^{2} \cdot 5^{4}$ 25.5.9536743164062500000000000000000000.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.100.10t6.a.a$2$ $ 2^{2} \cdot 5^{2}$ 25.5.9536743164062500000000000000000000.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.2500.10t6.a.c$2$ $ 2^{2} \cdot 5^{4}$ 25.5.9536743164062500000000000000000000.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.100.10t6.a.c$2$ $ 2^{2} \cdot 5^{2}$ 25.5.9536743164062500000000000000000000.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.100.10t6.a.d$2$ $ 2^{2} \cdot 5^{2}$ 25.5.9536743164062500000000000000000000.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.2500.5t2.a.b$2$ $ 2^{2} \cdot 5^{4}$ 5.1.6250000.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.100.10t6.a.b$2$ $ 2^{2} \cdot 5^{2}$ 25.5.9536743164062500000000000000000000.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.2500.10t6.a.b$2$ $ 2^{2} \cdot 5^{4}$ 25.5.9536743164062500000000000000000000.1 $C_5\times D_5$ (as 25T3) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.