Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 41*x^38 + 778*x^36 + 9065*x^34 + 72556*x^32 + 422839*x^30 + 1855505*x^28 + 6253910*x^26 + 16367625*x^24 + 33406879*x^22 + 53106978*x^20 + 65321146*x^18 + 61394717*x^16 + 43240794*x^14 + 22168369*x^12 + 7925719*x^10 + 1852081*x^8 + 255047*x^6 + 17190*x^4 + 360*x^2 + 1)
gp: K = bnfinit(y^40 + 41*y^38 + 778*y^36 + 9065*y^34 + 72556*y^32 + 422839*y^30 + 1855505*y^28 + 6253910*y^26 + 16367625*y^24 + 33406879*y^22 + 53106978*y^20 + 65321146*y^18 + 61394717*y^16 + 43240794*y^14 + 22168369*y^12 + 7925719*y^10 + 1852081*y^8 + 255047*y^6 + 17190*y^4 + 360*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 + 41*x^38 + 778*x^36 + 9065*x^34 + 72556*x^32 + 422839*x^30 + 1855505*x^28 + 6253910*x^26 + 16367625*x^24 + 33406879*x^22 + 53106978*x^20 + 65321146*x^18 + 61394717*x^16 + 43240794*x^14 + 22168369*x^12 + 7925719*x^10 + 1852081*x^8 + 255047*x^6 + 17190*x^4 + 360*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 + 41*x^38 + 778*x^36 + 9065*x^34 + 72556*x^32 + 422839*x^30 + 1855505*x^28 + 6253910*x^26 + 16367625*x^24 + 33406879*x^22 + 53106978*x^20 + 65321146*x^18 + 61394717*x^16 + 43240794*x^14 + 22168369*x^12 + 7925719*x^10 + 1852081*x^8 + 255047*x^6 + 17190*x^4 + 360*x^2 + 1)
\( x^{40} + 41 x^{38} + 778 x^{36} + 9065 x^{34} + 72556 x^{32} + 422839 x^{30} + 1855505 x^{28} + 6253910 x^{26} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(31654584865659568778929513407372752241664000000000000000000000000000000\)
\(\medspace = 2^{40}\cdot 5^{30}\cdot 11^{36}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(57.88\) | 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: | $2\cdot 5^{3/4}11^{9/10}\approx 57.87765351369302$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $40$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(220=2^{2}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(7,·)$, $\chi_{220}(9,·)$, $\chi_{220}(13,·)$, $\chi_{220}(17,·)$, $\chi_{220}(73,·)$, $\chi_{220}(89,·)$, $\chi_{220}(153,·)$, $\chi_{220}(69,·)$, $\chi_{220}(159,·)$, $\chi_{220}(167,·)$, $\chi_{220}(169,·)$, $\chi_{220}(43,·)$, $\chi_{220}(173,·)$, $\chi_{220}(49,·)$, $\chi_{220}(179,·)$, $\chi_{220}(181,·)$, $\chi_{220}(183,·)$, $\chi_{220}(57,·)$, $\chi_{220}(31,·)$, $\chi_{220}(63,·)$, $\chi_{220}(193,·)$, $\chi_{220}(197,·)$, $\chi_{220}(71,·)$, $\chi_{220}(201,·)$, $\chi_{220}(141,·)$, $\chi_{220}(81,·)$, $\chi_{220}(83,·)$, $\chi_{220}(87,·)$, $\chi_{220}(217,·)$, $\chi_{220}(91,·)$, $\chi_{220}(199,·)$, $\chi_{220}(59,·)$, $\chi_{220}(107,·)$, $\chi_{220}(111,·)$, $\chi_{220}(123,·)$, $\chi_{220}(117,·)$, $\chi_{220}(119,·)$, $\chi_{220}(191,·)$, $\chi_{220}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{524288}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{2}\times C_{85622}$, which has order $171244$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( a^{33} + 33 a^{31} + 495 a^{29} + 4466 a^{27} + 27027 a^{25} + 115830 a^{23} + 361790 a^{21} + 834900 a^{19} + 1427679 a^{17} + 1797818 a^{15} + 1641486 a^{13} + 1058147 a^{11} + 461879 a^{9} + 127864 a^{7} + 20119 a^{5} + 1441 a^{3} + 22 a \)
(order $4$)
| 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: |
$a^{22}+22a^{20}+209a^{18}+1122a^{16}+3740a^{14}+8008a^{12}+11011a^{10}+9438a^{8}+4719a^{6}+1210a^{4}+121a^{2}+2$, $a^{25}+25a^{23}+275a^{21}+1750a^{19}+7125a^{17}+19380a^{15}+35700a^{13}+44200a^{11}+35750a^{9}+17875a^{7}+5005a^{5}+650a^{3}+25a$, $a^{15}+15a^{13}+90a^{11}+275a^{9}+450a^{7}+378a^{5}+140a^{3}+15a$, $a^{38}+38a^{36}+665a^{34}+7106a^{32}+51831a^{30}+273265a^{28}+1075670a^{26}+3219750a^{24}+7393879a^{22}+13046957a^{20}+17600726a^{18}+17934897a^{16}+13523994a^{14}+7308094a^{12}+2691699a^{10}+622006a^{8}+76847a^{6}+3065a^{4}-140a^{2}-4$, $a^{15}+15a^{13}+90a^{11}+275a^{9}+450a^{7}+377a^{5}+135a^{3}+10a$, $a^{35}+35a^{33}+560a^{31}+5425a^{29}+35525a^{27}+166256a^{25}+573275a^{23}+1479775a^{21}+2876125a^{19}+4199000a^{17}+4556885a^{15}+3604525a^{13}+2013399a^{11}+755864a^{9}+176331a^{7}+22427a^{5}+1215a^{3}+9a$, $a^{38}+38a^{36}+665a^{34}+7106a^{32}+51832a^{30}+273295a^{28}+1076075a^{26}+3223000a^{24}+7411128a^{22}+13110691a^{20}+17768762a^{18}+18253545a^{16}+13956304a^{14}+7719986a^{12}+2957823a^{10}+731918a^{8}+103103a^{6}+6105a^{4}-11a^{2}-1$, $a^{35}+36a^{33}+593a^{31}+5920a^{29}+39991a^{27}+193284a^{25}+689130a^{23}+1841840a^{21}+3712775a^{19}+5633804a^{17}+6374082a^{15}+5281696a^{13}+3115658a^{11}+1253240a^{9}+321708a^{7}+47328a^{5}+3281a^{3}+68a$, $a^{33}+33a^{31}+495a^{29}+4466a^{27}+27026a^{25}+115805a^{23}+361515a^{21}+833150a^{19}+1420554a^{17}+1778438a^{15}+1605786a^{13}+1013948a^{11}+426140a^{9}+110033a^{7}+15191a^{5}+846a^{3}+8a$, $a^{6}+6a^{4}+9a^{2}+2$, $a^{23}+23a^{21}+230a^{19}+1311a^{17}+4692a^{15}+10948a^{13}+16744a^{11}+16445a^{9}+9867a^{7}+3289a^{5}+506a^{3}+23a$, $a^{9}+9a^{7}+27a^{5}+30a^{3}+9a$, $a^{37}+37a^{35}+629a^{33}+6512a^{31}+45880a^{29}+232841a^{27}+878787a^{25}+2510820a^{23}+5476185a^{21}+9126975a^{19}+11560835a^{17}+10994920a^{15}+7696444a^{13}+3848222a^{11}+1314610a^{9}+286824a^{7}+35853a^{5}+2109a^{3}+37a$, $a^{19}+19a^{17}+152a^{15}+665a^{13}+1729a^{11}+2717a^{9}+2508a^{7}+1254a^{5}+285a^{3}+19a$, $a^{27}+27a^{25}+324a^{23}+2277a^{21}+10395a^{19}+32319a^{17}+69768a^{15}+104652a^{13}+107406a^{11}+72930a^{9}+30888a^{7}+7371a^{5}+819a^{3}+27a$, $a^{24}+24a^{22}+252a^{20}+1520a^{18}+5814a^{16}+14688a^{14}+24752a^{12}+27456a^{10}+19305a^{8}+8008a^{6}+1716a^{4}+144a^{2}+3$, $a^{34}+34a^{32}+527a^{30}+4930a^{28}+31059a^{26}+139230a^{24}+457470a^{22}+1118260a^{20}+2042975a^{18}+2778446a^{16}+2778446a^{14}+1998724a^{12}+999362a^{10}+329460a^{8}+65892a^{6}+6936a^{4}+289a^{2}+2$, $a^{36}+36a^{34}+593a^{32}+5920a^{30}+39991a^{28}+193283a^{26}+689104a^{24}+1841542a^{22}+3710794a^{20}+5625348a^{18}+6349841a^{16}+5234263a^{14}+3052528a^{12}+1197306a^{10}+290092a^{8}+36729a^{6}+1430a^{4}-55a^{2}+1$, $a^{36}+36a^{34}+594a^{32}+5951a^{30}+40425a^{28}+196911a^{26}+709280a^{24}+1920270a^{22}+3932379a^{20}+6080855a^{18}+7034940a^{16}+5982605a^{14}+3636320a^{12}+1513501a^{10}+403711a^{8}+61798a^{6}+4345a^{4}+66a^{2}$
| 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: | \( 60790762850097.93 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 60790762850097.93 \cdot 171244}{4\cdot\sqrt{31654584865659568778929513407372752241664000000000000000000000000000000}}\cr\approx \mathstrut & 0.134515404426186 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^40 + 41*x^38 + 778*x^36 + 9065*x^34 + 72556*x^32 + 422839*x^30 + 1855505*x^28 + 6253910*x^26 + 16367625*x^24 + 33406879*x^22 + 53106978*x^20 + 65321146*x^18 + 61394717*x^16 + 43240794*x^14 + 22168369*x^12 + 7925719*x^10 + 1852081*x^8 + 255047*x^6 + 17190*x^4 + 360*x^2 + 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^40 + 41*x^38 + 778*x^36 + 9065*x^34 + 72556*x^32 + 422839*x^30 + 1855505*x^28 + 6253910*x^26 + 16367625*x^24 + 33406879*x^22 + 53106978*x^20 + 65321146*x^18 + 61394717*x^16 + 43240794*x^14 + 22168369*x^12 + 7925719*x^10 + 1852081*x^8 + 255047*x^6 + 17190*x^4 + 360*x^2 + 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^40 + 41*x^38 + 778*x^36 + 9065*x^34 + 72556*x^32 + 422839*x^30 + 1855505*x^28 + 6253910*x^26 + 16367625*x^24 + 33406879*x^22 + 53106978*x^20 + 65321146*x^18 + 61394717*x^16 + 43240794*x^14 + 22168369*x^12 + 7925719*x^10 + 1852081*x^8 + 255047*x^6 + 17190*x^4 + 360*x^2 + 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 + 41*x^38 + 778*x^36 + 9065*x^34 + 72556*x^32 + 422839*x^30 + 1855505*x^28 + 6253910*x^26 + 16367625*x^24 + 33406879*x^22 + 53106978*x^20 + 65321146*x^18 + 61394717*x^16 + 43240794*x^14 + 22168369*x^12 + 7925719*x^10 + 1852081*x^8 + 255047*x^6 + 17190*x^4 + 360*x^2 + 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2\times C_{20}$ (as 40T2):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| An abelian group of order 40 |
| The 40 conjugacy class representatives for $C_2\times C_{20}$ |
| Character table for $C_2\times C_{20}$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-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 | $20^{2}$ | R | $20^{2}$ | R | $20^{2}$ | $20^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{10}$ | ${\href{/padicField/29.5.0.1}{5} }^{8}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $40$ | $2$ | $20$ | $40$ | |||
|
\(5\)
| Deg $20$ | $4$ | $5$ | $15$ | |||
| Deg $20$ | $4$ | $5$ | $15$ | ||||
|
\(11\)
| 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |
| 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |