Normalized defining polynomial

\( x^{26} - x^{25} - 25 x^{24} + 24 x^{23} + 276 x^{22} - 253 x^{21} - 1771 x^{20} + 1540 x^{19} + 7315 x^{18} + \cdots - 1 \) x^26 - x^25 - 25*x^24 + 24*x^23 + 276*x^22 - 253*x^21 - 1771*x^20 + 1540*x^19 + 7315*x^18 - 5985*x^17 - 20349*x^16 + 15504*x^15 + 38760*x^14 - 27132*x^13 - 50388*x^12 + 31824*x^11 + 43758*x^10 - 24310*x^9 - 24310*x^8 + 11440*x^7 + 8008*x^6 - 3003*x^5 - 1365*x^4 + 364*x^3 + 91*x^2 - 13*x - 1

Invariants

Degree:		$26$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[26, 0]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(12790771483610519443342791266451996229460693\) 12790771483610519443342791266451996229460693 \(\medspace = 53^{25}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(45.49\)	sage: (K.disc().abs())^(1./K.degree())   gp: abs(K.disc)^(1/poldegree(K.pol))   magma: Abs(Discriminant(OK))^(1/Degree(K));   oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:		$53^{25/26}\approx 45.494360736359816$
Ramified primes:		\(53\) 53	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{53}) \)
$\card{ \Gal(K/\Q) }$:		$26$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:		\(53\)
Dirichlet character group:		$\lbrace$$\chi_{53}(1,·)$, $\chi_{53}(4,·)$, $\chi_{53}(6,·)$, $\chi_{53}(7,·)$, $\chi_{53}(9,·)$, $\chi_{53}(10,·)$, $\chi_{53}(11,·)$, $\chi_{53}(13,·)$, $\chi_{53}(15,·)$, $\chi_{53}(16,·)$, $\chi_{53}(17,·)$, $\chi_{53}(24,·)$, $\chi_{53}(25,·)$, $\chi_{53}(28,·)$, $\chi_{53}(29,·)$, $\chi_{53}(36,·)$, $\chi_{53}(37,·)$, $\chi_{53}(38,·)$, $\chi_{53}(40,·)$, $\chi_{53}(42,·)$, $\chi_{53}(43,·)$, $\chi_{53}(44,·)$, $\chi_{53}(46,·)$, $\chi_{53}(47,·)$, $\chi_{53}(49,·)$, $\chi_{53}(52,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$ 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, a^19, a^20, a^21, a^22, a^23, a^24, a^25

Class group and class number

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

Unit group

Rank:		$25$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -1  (order $2$)	sage: UK.torsion_generator()   gp: K.tu[2]   magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);   oscar: torsion_units_generator(OK)
Fundamental units:		$a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-2$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{5}-5a^{3}+5a$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5732a^{11}-a^{10}+6996a^{9}+10a^{8}-5104a^{7}-35a^{6}+2002a^{5}+50a^{4}-330a^{3}-25a^{2}+10a+1$, $a^{25}-25a^{23}-a^{22}+275a^{21}+22a^{20}-1750a^{19}-209a^{18}+7125a^{17}+1122a^{16}-19380a^{15}-3740a^{14}+35700a^{13}+8008a^{12}-44200a^{11}-11011a^{10}+35750a^{9}+9438a^{8}-17875a^{7}-4719a^{6}+5005a^{5}+1210a^{4}-650a^{3}-121a^{2}+25a+2$, $a^{3}-3a$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}-a^{5}+1716a^{4}+5a^{3}-144a^{2}-5a+3$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24753a^{12}-27468a^{10}+19359a^{8}-8120a^{6}+1821a^{4}-180a^{2}+5$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+2$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}-a^{16}+4692a^{15}+15a^{14}-10948a^{13}-90a^{12}+16744a^{11}+275a^{10}-16444a^{9}-450a^{8}+9859a^{7}+378a^{6}-3269a^{5}-140a^{4}+490a^{3}+15a^{2}-21a-1$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a$, $a^{5}-5a^{3}+5a-1$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+3$, $a^{4}-4a^{2}+3$, $a^{5}-4a^{3}+3a$, $a^{24}-24a^{22}+252a^{20}-a^{19}-1520a^{18}+19a^{17}+5814a^{16}-152a^{15}-14687a^{14}+665a^{13}+24738a^{12}-1729a^{11}-27378a^{10}+2717a^{9}+19085a^{8}-2508a^{7}-7679a^{6}+1253a^{5}+1470a^{4}-280a^{3}-70a^{2}+14a-1$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+2$, $a^{24}-24a^{22}+252a^{20}-a^{19}-1520a^{18}+19a^{17}+5814a^{16}-152a^{15}-14688a^{14}+665a^{13}+24752a^{12}-1729a^{11}-27456a^{10}+2717a^{9}+19305a^{8}-2508a^{7}-8008a^{6}+1253a^{5}+1716a^{4}-280a^{3}-144a^{2}+14a+3$, $a^{20}-a^{19}-19a^{18}+18a^{17}+153a^{16}-136a^{15}-681a^{14}+561a^{13}+1833a^{12}-1377a^{11}-3069a^{10}+2057a^{9}+3168a^{8}-1836a^{7}-1926a^{6}+918a^{5}+621a^{4}-221a^{3}-82a^{2}+16a+1$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+3$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+3$ a^14 - 14*a^12 + 77*a^10 - 210*a^8 + 294*a^6 - 196*a^4 + 49*a^2 - 2, a^6 - 6*a^4 + 9*a^2 - 2, a^12 - 12*a^10 + 54*a^8 - 112*a^6 + 105*a^4 - 36*a^2 + 2, a^24 - 24*a^22 + 252*a^20 - 1520*a^18 + 5814*a^16 - 14688*a^14 + 24752*a^12 - 27456*a^10 + 19305*a^8 - 8008*a^6 + 1716*a^4 - 144*a^2 + 2, a^5 - 5*a^3 + 5*a, a^10 - 10*a^8 + 35*a^6 - 50*a^4 + 25*a^2 - 2, a^16 - 16*a^14 + 104*a^12 - 352*a^10 + 660*a^8 - 672*a^6 + 336*a^4 - 64*a^2 + 2, a^21 - 21*a^19 + 189*a^17 - 952*a^15 + 2940*a^13 - 5732*a^11 - a^10 + 6996*a^9 + 10*a^8 - 5104*a^7 - 35*a^6 + 2002*a^5 + 50*a^4 - 330*a^3 - 25*a^2 + 10*a + 1, a^25 - 25*a^23 - a^22 + 275*a^21 + 22*a^20 - 1750*a^19 - 209*a^18 + 7125*a^17 + 1122*a^16 - 19380*a^15 - 3740*a^14 + 35700*a^13 + 8008*a^12 - 44200*a^11 - 11011*a^10 + 35750*a^9 + 9438*a^8 - 17875*a^7 - 4719*a^6 + 5005*a^5 + 1210*a^4 - 650*a^3 - 121*a^2 + 25*a + 2, a^3 - 3*a, a^24 - 24*a^22 + 252*a^20 - 1520*a^18 + 5814*a^16 - 14688*a^14 + 24752*a^12 - 27456*a^10 + 19305*a^8 - 8008*a^6 - a^5 + 1716*a^4 + 5*a^3 - 144*a^2 - 5*a + 3, a^24 - 24*a^22 + 252*a^20 - 1520*a^18 + 5814*a^16 - 14688*a^14 + 24753*a^12 - 27468*a^10 + 19359*a^8 - 8120*a^6 + 1821*a^4 - 180*a^2 + 5, a^8 - 8*a^6 + 20*a^4 - 16*a^2 + 2, a^23 - 23*a^21 + 230*a^19 - 1311*a^17 - a^16 + 4692*a^15 + 15*a^14 - 10948*a^13 - 90*a^12 + 16744*a^11 + 275*a^10 - 16444*a^9 - 450*a^8 + 9859*a^7 + 378*a^6 - 3269*a^5 - 140*a^4 + 490*a^3 + 15*a^2 - 21*a - 1, a^21 - 21*a^19 + 189*a^17 - 952*a^15 + 2940*a^13 - 5733*a^11 + 7007*a^9 - 5148*a^7 + 2079*a^5 - 385*a^3 + 21*a, a^5 - 5*a^3 + 5*a - 1, a^8 - 8*a^6 + 20*a^4 - 16*a^2 + 3, a^4 - 4*a^2 + 3, a^5 - 4*a^3 + 3*a, a^24 - 24*a^22 + 252*a^20 - a^19 - 1520*a^18 + 19*a^17 + 5814*a^16 - 152*a^15 - 14687*a^14 + 665*a^13 + 24738*a^12 - 1729*a^11 - 27378*a^10 + 2717*a^9 + 19085*a^8 - 2508*a^7 - 7679*a^6 + 1253*a^5 + 1470*a^4 - 280*a^3 - 70*a^2 + 14*a - 1, a^20 - 20*a^18 + 170*a^16 - 800*a^14 + 2275*a^12 - 4004*a^10 + 4290*a^8 - 2640*a^6 + 825*a^4 - 100*a^2 + 2, a^24 - 24*a^22 + 252*a^20 - a^19 - 1520*a^18 + 19*a^17 + 5814*a^16 - 152*a^15 - 14688*a^14 + 665*a^13 + 24752*a^12 - 1729*a^11 - 27456*a^10 + 2717*a^9 + 19305*a^8 - 2508*a^7 - 8008*a^6 + 1253*a^5 + 1716*a^4 - 280*a^3 - 144*a^2 + 14*a + 3, a^20 - a^19 - 19*a^18 + 18*a^17 + 153*a^16 - 136*a^15 - 681*a^14 + 561*a^13 + 1833*a^12 - 1377*a^11 - 3069*a^10 + 2057*a^9 + 3168*a^8 - 1836*a^7 - 1926*a^6 + 918*a^5 + 621*a^4 - 221*a^3 - 82*a^2 + 16*a + 1, a^12 - 12*a^10 + 54*a^8 - 112*a^6 + 105*a^4 - 36*a^2 + 3, a^24 - 24*a^22 + 252*a^20 - 1520*a^18 + 5814*a^16 - 14688*a^14 + 24752*a^12 - 27456*a^10 + 19305*a^8 - 8008*a^6 + 1716*a^4 - 144*a^2 + 3 (assuming GRH)	sage: UK.fundamental_units()   gp: K.fu   magma: [K|fUK(g): g in Generators(UK)];   oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:		\( 14915851505236.459 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{26}\cdot(2\pi)^{0}\cdot 14915851505236.459 \cdot 1}{2\cdot\sqrt{12790771483610519443342791266451996229460693}}\cr\approx \mathstrut & 0.139942482145618 \end{aligned}\] (assuming GRH)

Galois group

$C_{26}$ (as 26T1):

Intermediate fields

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

Frobenius cycle types

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

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(53\) 53		Deg $26$	$26$	$1$	$25$