LMFDB - Number field 12.0.161428875495388931828125.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881)

gp: K = bnfinit(y^12 - y^11 + 66*y^10 - 170*y^9 + 1249*y^8 - 3446*y^7 + 12689*y^6 - 57227*y^5 + 67939*y^4 - 83461*y^3 + 539865*y^2 - 552644*y + 152881, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881)

$x^{12} - x^{11} + 66 x^{10} - 170 x^{9} + 1249 x^{8} - 3446 x^{7} + 12689 x^{6} - 57227 x^{5} + \cdots + 152881$ x^12 - x^11 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$12$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 6]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$161428875495388931828125$ 161428875495388931828125 $\medspace = 5^{6}\cdot 7^{8}\cdot 13^{11}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$85.90$	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:	$5^{1/2}7^{2/3}13^{11/12}\approx 85.90104262646453$
Ramified primes:	$5$ , $7$ , $13$ 5, 7, 13	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{13})$
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$ :	$C_{12}$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$ .
Conductor:	$455=5\cdot 7\cdot 13$
Dirichlet character group:	$\lbrace$ $\chi_{455}(256,·)$ , $\chi_{455}(1,·)$ , $\chi_{455}(99,·)$ , $\chi_{455}(214,·)$ , $\chi_{455}(296,·)$ , $\chi_{455}(239,·)$ , $\chi_{455}(16,·)$ , $\chi_{455}(246,·)$ , $\chi_{455}(184,·)$ , $\chi_{455}(186,·)$ , $\chi_{455}(219,·)$ , $\chi_{455}(319,·)$ $\rbrace$
This is a CM field.
Reflex fields:	4.0.54925.1 $^{2}$ , 12.0.161428875495388931828125.1 $^{30}$

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

$1$ , $a$ , $a^{2}$ , $a^{3}$ , $a^{4}$ , $a^{5}$ , $a^{6}$ , $a^{7}$ , $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$ , $\frac{1}{782}a^{9}-\frac{71}{782}a^{8}-\frac{4}{23}a^{7}+\frac{121}{782}a^{6}-\frac{5}{46}a^{5}+\frac{337}{782}a^{4}-\frac{12}{391}a^{3}+\frac{89}{782}a^{2}+\frac{227}{782}a$ , $\frac{1}{782}a^{10}-\frac{47}{391}a^{8}+\frac{120}{391}a^{7}-\frac{48}{391}a^{6}+\frac{167}{782}a^{5}+\frac{26}{391}a^{4}+\frac{10}{23}a^{3}+\frac{145}{391}a^{2}+\frac{43}{391}a-\frac{1}{2}$ , $\frac{1}{16\!\cdots\!86}a^{11}-\frac{44\!\cdots\!53}{99\!\cdots\!58}a^{10}+\frac{82\!\cdots\!88}{84\!\cdots\!93}a^{9}-\frac{86\!\cdots\!50}{84\!\cdots\!93}a^{8}+\frac{20\!\cdots\!61}{84\!\cdots\!93}a^{7}-\frac{82\!\cdots\!33}{16\!\cdots\!86}a^{6}+\frac{84\!\cdots\!87}{16\!\cdots\!86}a^{5}+\frac{42\!\cdots\!94}{84\!\cdots\!93}a^{4}+\frac{79\!\cdots\!65}{49\!\cdots\!29}a^{3}-\frac{41\!\cdots\!76}{84\!\cdots\!93}a^{2}-\frac{34\!\cdots\!05}{73\!\cdots\!82}a-\frac{21\!\cdots\!01}{18\!\cdots\!02}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/2*a^8 - 1/2*a^7 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a - 1/2, 1/782*a^9 - 71/782*a^8 - 4/23*a^7 + 121/782*a^6 - 5/46*a^5 + 337/782*a^4 - 12/391*a^3 + 89/782*a^2 + 227/782*a, 1/782*a^10 - 47/391*a^8 + 120/391*a^7 - 48/391*a^6 + 167/782*a^5 + 26/391*a^4 + 10/23*a^3 + 145/391*a^2 + 43/391*a - 1/2, 1/16917891751036173218345386*a^11 - 440397629297584342553/995170103002127836373258*a^10 + 828822223521701659088/8458945875518086609172693*a^9 - 860895265422671217265450/8458945875518086609172693*a^8 + 2085269205725185848776861/8458945875518086609172693*a^7 - 8203583994490899035321333/16917891751036173218345386*a^6 + 8438528897301077950992087/16917891751036173218345386*a^5 + 4207716131436924294687794/8458945875518086609172693*a^4 + 79548680493576954258265/497585051501063918186629*a^3 - 4179706426594888115150476/8458945875518086609172693*a^2 - 343812937400989344929705/735560510914616226884582*a - 218345787822910340601/1881228928170373981802

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

$C_{13}\times C_{78}$ , which has order $1014$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Relative class number: $338$ (assuming GRH)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$5$	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:	$\frac{553359443973002}{65\!\cdots\!57}a^{11}-\frac{32\!\cdots\!61}{65\!\cdots\!57}a^{10}+\frac{24\!\cdots\!78}{38\!\cdots\!21}a^{9}-\frac{15\!\cdots\!16}{38\!\cdots\!21}a^{8}+\frac{13\!\cdots\!46}{65\!\cdots\!57}a^{7}-\frac{53\!\cdots\!16}{65\!\cdots\!57}a^{6}+\frac{18\!\cdots\!08}{65\!\cdots\!57}a^{5}-\frac{64\!\cdots\!04}{65\!\cdots\!57}a^{4}+\frac{17\!\cdots\!06}{65\!\cdots\!57}a^{3}-\frac{25\!\cdots\!78}{65\!\cdots\!57}a^{2}+\frac{70\!\cdots\!38}{28\!\cdots\!59}a-\frac{59\!\cdots\!23}{16\!\cdots\!27}$ , $\frac{73\!\cdots\!77}{16\!\cdots\!86}a^{11}+\frac{29\!\cdots\!47}{16\!\cdots\!86}a^{10}+\frac{30\!\cdots\!73}{16\!\cdots\!86}a^{9}+\frac{93\!\cdots\!09}{84\!\cdots\!93}a^{8}-\frac{33\!\cdots\!59}{16\!\cdots\!86}a^{7}+\frac{97\!\cdots\!90}{49\!\cdots\!29}a^{6}-\frac{74\!\cdots\!47}{16\!\cdots\!86}a^{5}+\frac{14\!\cdots\!14}{84\!\cdots\!93}a^{4}-\frac{14\!\cdots\!45}{16\!\cdots\!86}a^{3}+\frac{10\!\cdots\!55}{16\!\cdots\!86}a^{2}+\frac{10\!\cdots\!61}{73\!\cdots\!82}a+\frac{47\!\cdots\!29}{94\!\cdots\!01}$ , $\frac{80\!\cdots\!87}{16\!\cdots\!86}a^{11}+\frac{19\!\cdots\!21}{16\!\cdots\!86}a^{10}+\frac{54\!\cdots\!61}{16\!\cdots\!86}a^{9}+\frac{23\!\cdots\!32}{84\!\cdots\!93}a^{8}+\frac{81\!\cdots\!47}{16\!\cdots\!86}a^{7}+\frac{16\!\cdots\!87}{84\!\cdots\!93}a^{6}+\frac{54\!\cdots\!47}{16\!\cdots\!86}a^{5}-\frac{57\!\cdots\!04}{49\!\cdots\!29}a^{4}-\frac{64\!\cdots\!35}{16\!\cdots\!86}a^{3}-\frac{79\!\cdots\!65}{16\!\cdots\!86}a^{2}+\frac{70\!\cdots\!75}{73\!\cdots\!82}a+\frac{31\!\cdots\!17}{94\!\cdots\!01}$ , $\frac{33\!\cdots\!67}{99\!\cdots\!58}a^{11}+\frac{34\!\cdots\!37}{16\!\cdots\!86}a^{10}+\frac{19\!\cdots\!54}{84\!\cdots\!93}a^{9}+\frac{83\!\cdots\!41}{84\!\cdots\!93}a^{8}+\frac{17\!\cdots\!09}{84\!\cdots\!93}a^{7}+\frac{23\!\cdots\!61}{16\!\cdots\!86}a^{6}-\frac{16\!\cdots\!43}{16\!\cdots\!86}a^{5}+\frac{29\!\cdots\!68}{84\!\cdots\!93}a^{4}-\frac{69\!\cdots\!46}{84\!\cdots\!93}a^{3}+\frac{54\!\cdots\!89}{49\!\cdots\!29}a^{2}+\frac{46\!\cdots\!27}{73\!\cdots\!82}a+\frac{18\!\cdots\!05}{18\!\cdots\!02}$ , $\frac{11\!\cdots\!21}{16\!\cdots\!86}a^{11}+\frac{12\!\cdots\!69}{16\!\cdots\!86}a^{10}+\frac{33\!\cdots\!63}{84\!\cdots\!93}a^{9}+\frac{17\!\cdots\!67}{84\!\cdots\!93}a^{8}+\frac{25\!\cdots\!19}{84\!\cdots\!93}a^{7}+\frac{17\!\cdots\!61}{16\!\cdots\!86}a^{6}-\frac{21\!\cdots\!53}{16\!\cdots\!86}a^{5}-\frac{13\!\cdots\!52}{84\!\cdots\!93}a^{4}-\frac{10\!\cdots\!38}{84\!\cdots\!93}a^{3}-\frac{22\!\cdots\!01}{84\!\cdots\!93}a^{2}+\frac{11\!\cdots\!41}{73\!\cdots\!82}a+\frac{11\!\cdots\!39}{18\!\cdots\!02}$ 553359443973002/653250897792731995457a^11 - 3223944337855161/653250897792731995457a^10 + 2465028414647178/38426523399572470321a^9 - 15867715156112716/38426523399572470321a^8 + 1318089094300401646/653250897792731995457a^7 - 5379266361210799616/653250897792731995457a^6 + 18426277703679171808/653250897792731995457a^5 - 64484827945434136704/653250897792731995457a^4 + 175539436106630101106/653250897792731995457a^3 - 258364937428634330878/653250897792731995457a^2 + 7073916561310525138/28402212947510086759a - 597149733943479423/1670718408677063927, 7366772862689083777/16917891751036173218345386a^11 + 297021219645994795347/16917891751036173218345386a^10 + 305472017455523116573/16917891751036173218345386a^9 + 9335591271944904694509/8458945875518086609172693a^8 - 33569873595504716143259/16917891751036173218345386a^7 + 9735370644504548891990/497585051501063918186629a^6 - 748802113895942913357547/16917891751036173218345386a^5 + 1440817068575813127052614/8458945875518086609172693a^4 - 14554555612597284758914945/16917891751036173218345386a^3 + 10652348882978006119066755/16917891751036173218345386a^2 + 1045654081860159055361/735560510914616226884582a + 4741463486219343156829/940614464085186990901, 80383421623106090087/16917891751036173218345386a^11 + 195577717310728336621/16917891751036173218345386a^10 + 5458869144211788223861/16917891751036173218345386a^9 + 2353666630539503183732/8458945875518086609172693a^8 + 81942693481477660241347/16917891751036173218345386a^7 + 16273358168206198081387/8458945875518086609172693a^6 + 541709608316415744638347/16917891751036173218345386a^5 - 57709994159775966394004/497585051501063918186629a^4 - 6490740160017095279932135/16917891751036173218345386a^3 - 7949178441240147022371265/16917891751036173218345386a^2 + 704330227361842146105375/735560510914616226884582a + 3158773900117771267217/940614464085186990901, 3392968267538765867/995170103002127836373258a^11 + 345327179062767406437/16917891751036173218345386a^10 + 1943871489928138413154/8458945875518086609172693a^9 + 8301757180541564055041/8458945875518086609172693a^8 + 17027337936542021218909/8458945875518086609172693a^7 + 239426892117590204347861/16917891751036173218345386a^6 - 160854125336786954647843/16917891751036173218345386a^5 + 291417903338141612298668/8458945875518086609172693a^4 - 6980662703928215864474746/8458945875518086609172693a^3 + 54692704978392549419489/497585051501063918186629a^2 + 466028280940589648500227/735560510914616226884582a + 18065750319643044435305/1881228928170373981802, 11971001725490685221/16917891751036173218345386a^11 + 12837016150612252569/16917891751036173218345386a^10 + 337509359219593077463/8458945875518086609172693a^9 + 1705408570603957467/8458945875518086609172693a^8 + 2595789533179940763719/8458945875518086609172693a^7 + 17972619500270200823061/16917891751036173218345386a^6 - 21127544418524647182153/16917891751036173218345386a^5 - 131206653127654228995652/8458945875518086609172693a^4 - 10642845049658420755438/8458945875518086609172693a^3 - 22274596521528643239201/8458945875518086609172693a^2 + 11255956989059192492841/735560510914616226884582*a + 1130755093389572900339/1881228928170373981802 (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:	$6176.96689838$ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 6176.96689838 \cdot 1014}{2\cdot\sqrt{161428875495388931828125}}\cr\approx \mathstrut & 0.479591892500 \end{aligned}$ (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881)

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^12 - x^11 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881, 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(Rationals()); K<a> := NumberField(x^12 - x^11 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881);

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^12 - x^11 + 66*x^10 - 170*x^9 + 1249*x^8 - 3446*x^7 + 12689*x^6 - 57227*x^5 + 67939*x^4 - 83461*x^3 + 539865*x^2 - 552644*x + 152881);

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

$C_{12}$ (as 12T1):

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)

A cyclic group of order 12

The 12 conjugacy class representatives for

C_{12}

Character table for

C_{12}

Intermediate fields

\Q(\sqrt{13})

, 3.3.8281.1, 4.0.54925.1, 6.6.891474493.1

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	${\href{/padicField/2.4.0.1}{4} }^{3}$	${\href{/padicField/3.6.0.1}{6} }^{2}$	R	R	${\href{/padicField/11.12.0.1}{12} }$	R	${\href{/padicField/17.1.0.1}{1} }^{12}$	${\href{/padicField/19.12.0.1}{12} }$	${\href{/padicField/23.1.0.1}{1} }^{12}$	${\href{/padicField/29.3.0.1}{3} }^{4}$	${\href{/padicField/31.12.0.1}{12} }$	${\href{/padicField/37.4.0.1}{4} }^{3}$	${\href{/padicField/41.12.0.1}{12} }$	${\href{/padicField/43.3.0.1}{3} }^{4}$	${\href{/padicField/47.12.0.1}{12} }$	${\href{/padicField/53.6.0.1}{6} }^{2}$	${\href{/padicField/59.4.0.1}{4} }^{3}$

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
$5$ 5	5.6.2.6a1.1	$x^{12} + 2 x^{10} + 8 x^{9} + 3 x^{8} + 8 x^{7} + 22 x^{6} + 8 x^{5} + 5 x^{4} + 16 x^{3} + 4 x^{2} + 5 x + 4$	$2$	$6$	$6$	$C_{12}$	$[\ ]_{2}^{6}$
$7$ 7	7.4.3.8a1.1	$x^{12} + 15 x^{10} + 12 x^{9} + 84 x^{8} + 120 x^{7} + 263 x^{6} + 372 x^{5} + 492 x^{4} + 424 x^{3} + 286 x^{2} + 108 x + 27$	$3$	$4$	$8$	$C_{12}$	$[\ ]_{3}^{4}$
$13$ 13	13.1.12.11a1.12	$x^{12} + 156$	$12$	$1$	$11$	$C_{12}$	$[\ ]_{12}$

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Normalized defining polynomial

Invariants

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

Class group and class number

Unit group

Class number formula

Galois group

Intermediate fields

Frobenius cycle types

Local algebras for ramified primes

Spectrum of ring of integers

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	12.0.161...125.1

Degree	$12$
Signature	$[0, 6]$
Discriminant	$1.614\times 10^{23}$
Root discriminant	$85.90$
Ramified primes	$5,7,13$
Class number	$1014$ (GRH)
Class group	[13, 78] (GRH)
Galois group	$C_{12}$ (as 12T1)

Properties

Related objects

Downloads

Learn more

Integral basis (with respect to field generator aaa)

Class group and class number

Local algebras for ramified primes

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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