Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - x^32 - 32*x^31 + 31*x^30 + 465*x^29 - 435*x^28 - 4060*x^27 + 3654*x^26 + 23751*x^25 - 20475*x^24 - 98280*x^23 + 80730*x^22 + 296010*x^21 - 230230*x^20 - 657800*x^19 + 480700*x^18 + 1081575*x^17 - 735471*x^16 - 1307504*x^15 + 817190*x^14 + 1144066*x^13 - 646646*x^12 - 705432*x^11 + 352716*x^10 + 293930*x^9 - 125970*x^8 - 77520*x^7 + 27132*x^6 + 11628*x^5 - 3060*x^4 - 816*x^3 + 136*x^2 + 17*x - 1)
gp: K = bnfinit(y^33 - y^32 - 32*y^31 + 31*y^30 + 465*y^29 - 435*y^28 - 4060*y^27 + 3654*y^26 + 23751*y^25 - 20475*y^24 - 98280*y^23 + 80730*y^22 + 296010*y^21 - 230230*y^20 - 657800*y^19 + 480700*y^18 + 1081575*y^17 - 735471*y^16 - 1307504*y^15 + 817190*y^14 + 1144066*y^13 - 646646*y^12 - 705432*y^11 + 352716*y^10 + 293930*y^9 - 125970*y^8 - 77520*y^7 + 27132*y^6 + 11628*y^5 - 3060*y^4 - 816*y^3 + 136*y^2 + 17*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^33 - x^32 - 32*x^31 + 31*x^30 + 465*x^29 - 435*x^28 - 4060*x^27 + 3654*x^26 + 23751*x^25 - 20475*x^24 - 98280*x^23 + 80730*x^22 + 296010*x^21 - 230230*x^20 - 657800*x^19 + 480700*x^18 + 1081575*x^17 - 735471*x^16 - 1307504*x^15 + 817190*x^14 + 1144066*x^13 - 646646*x^12 - 705432*x^11 + 352716*x^10 + 293930*x^9 - 125970*x^8 - 77520*x^7 + 27132*x^6 + 11628*x^5 - 3060*x^4 - 816*x^3 + 136*x^2 + 17*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^33 - x^32 - 32*x^31 + 31*x^30 + 465*x^29 - 435*x^28 - 4060*x^27 + 3654*x^26 + 23751*x^25 - 20475*x^24 - 98280*x^23 + 80730*x^22 + 296010*x^21 - 230230*x^20 - 657800*x^19 + 480700*x^18 + 1081575*x^17 - 735471*x^16 - 1307504*x^15 + 817190*x^14 + 1144066*x^13 - 646646*x^12 - 705432*x^11 + 352716*x^10 + 293930*x^9 - 125970*x^8 - 77520*x^7 + 27132*x^6 + 11628*x^5 - 3060*x^4 - 816*x^3 + 136*x^2 + 17*x - 1)
\( x^{33} - x^{32} - 32 x^{31} + 31 x^{30} + 465 x^{29} - 435 x^{28} - 4060 x^{27} + 3654 x^{26} + 23751 x^{25} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $33$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[33, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(27189028279553414235049966267283185807800188603627566700161\)
\(\medspace = 67^{32}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(58.98\) | 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: | $67^{32/33}\approx 58.98467660010086$ | ||
| Ramified primes: |
\(67\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $33$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(67\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{67}(1,·)$, $\chi_{67}(4,·)$, $\chi_{67}(6,·)$, $\chi_{67}(9,·)$, $\chi_{67}(10,·)$, $\chi_{67}(14,·)$, $\chi_{67}(15,·)$, $\chi_{67}(16,·)$, $\chi_{67}(17,·)$, $\chi_{67}(19,·)$, $\chi_{67}(21,·)$, $\chi_{67}(22,·)$, $\chi_{67}(23,·)$, $\chi_{67}(24,·)$, $\chi_{67}(25,·)$, $\chi_{67}(26,·)$, $\chi_{67}(29,·)$, $\chi_{67}(33,·)$, $\chi_{67}(35,·)$, $\chi_{67}(36,·)$, $\chi_{67}(37,·)$, $\chi_{67}(39,·)$, $\chi_{67}(40,·)$, $\chi_{67}(47,·)$, $\chi_{67}(49,·)$, $\chi_{67}(54,·)$, $\chi_{67}(55,·)$, $\chi_{67}(56,·)$, $\chi_{67}(59,·)$, $\chi_{67}(60,·)$, $\chi_{67}(62,·)$, $\chi_{67}(64,·)$, $\chi_{67}(65,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$
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
Trivial group, which has order $1$ (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: | $32$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(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: |
$a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}-1254a^{5}+285a^{3}-19a$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68952a^{12}+63206a^{10}-37180a^{8}+13013a^{6}-2366a^{4}+169a^{2}-2$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17251a^{22}-63778a^{20}+168454a^{18}-320892a^{16}-a^{15}+439790a^{14}+15a^{13}-427908a^{12}-90a^{11}+288145a^{10}+275a^{9}-128778a^{8}-450a^{7}+35659a^{6}+378a^{5}-5410a^{4}-140a^{3}+346a^{2}+15a-3$, $a^{3}-3a-1$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+5440a^{4}-256a^{2}+2$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{30}-30a^{28}+405a^{26}-a^{25}-3250a^{24}+25a^{23}+17250a^{22}-275a^{21}-63756a^{20}+1750a^{19}+168246a^{18}-7125a^{17}-319788a^{16}+19380a^{15}+436185a^{14}-35700a^{13}-420446a^{12}+44200a^{11}+278421a^{10}-35750a^{9}-121122a^{8}+17875a^{7}+32327a^{6}-5005a^{5}-4746a^{4}+650a^{3}+315a^{2}-25a-6$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-a^{18}-32319a^{17}+18a^{16}+69768a^{15}-135a^{14}-104652a^{13}+546a^{12}+107406a^{11}-1287a^{10}-72929a^{9}+1782a^{8}+30879a^{7}-1386a^{6}-7344a^{5}+540a^{4}+789a^{3}-81a^{2}-18a+1$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-a^{20}-51359a^{19}+20a^{18}+127281a^{17}-170a^{16}-224808a^{15}+800a^{14}+281010a^{13}-2275a^{12}-243542a^{11}+4004a^{10}+140999a^{9}-4290a^{8}-51281a^{7}+2640a^{6}+10583a^{5}-825a^{4}-1045a^{3}+100a^{2}+38a-3$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17251a^{22}-63778a^{20}+168454a^{18}-320892a^{16}-a^{15}+439790a^{14}+15a^{13}-427908a^{12}-90a^{11}+288145a^{10}+275a^{9}-128778a^{8}-451a^{7}+35659a^{6}+385a^{5}-5410a^{4}-154a^{3}+346a^{2}+22a-3$, $a^{29}-28a^{27}+350a^{25}-2576a^{23}+12397a^{21}-a^{20}-40964a^{19}+19a^{18}+94962a^{17}-152a^{16}-155040a^{15}+665a^{14}+176358a^{13}-1729a^{12}-136135a^{11}+2717a^{10}+68058a^{9}-2508a^{8}-20349a^{7}+1254a^{6}+3135a^{5}-285a^{4}-171a^{3}+18a^{2}+1$, $a^{31}-31a^{29}+435a^{27}-3654a^{25}+20474a^{23}-a^{22}-80707a^{21}+22a^{20}+229999a^{19}-210a^{18}-479370a^{17}+1140a^{16}+730626a^{15}-3875a^{14}-805561a^{13}+8554a^{12}+628069a^{11}-12298a^{10}-333202a^{9}+11220a^{8}+112935a^{7}-6105a^{6}-21918a^{5}+1750a^{4}+1938a^{3}-202a^{2}-36a+3$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7126a^{17}-19397a^{15}+35819a^{13}-44642a^{11}+36685a^{9}-18997a^{7}+5719a^{5}-854a^{3}+42a-1$, $a^{7}-7a^{5}+14a^{3}-7a-1$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+30940a^{6}-4200a^{4}+225a^{2}-1$, $a^{27}-27a^{25}-a^{24}+324a^{23}+24a^{22}-2277a^{21}-252a^{20}+10395a^{19}+1520a^{18}-32319a^{17}-5814a^{16}+69768a^{15}+14688a^{14}-104652a^{13}-24752a^{12}+107406a^{11}+27456a^{10}-72930a^{9}-19306a^{8}+30888a^{7}+8016a^{6}-7371a^{5}-1736a^{4}+819a^{3}+160a^{2}-27a-4$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}-a^{15}+436050a^{14}+15a^{13}-419900a^{12}-90a^{11}+277134a^{10}+275a^{9}-119340a^{8}-450a^{7}+30940a^{6}+378a^{5}-4200a^{4}-140a^{3}+225a^{2}+15a-1$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}-700910a^{13}-a^{12}+520676a^{11}+12a^{10}-260338a^{9}-54a^{8}+82212a^{7}+112a^{6}-14756a^{5}-105a^{4}+1240a^{3}+36a^{2}-31a-2$, $a^{5}-5a^{3}+5a-1$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-1$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a-1$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-1$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a-1$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a-1$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+30940a^{6}-4200a^{4}+225a^{2}-2$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140999a^{9}-51281a^{7}+10583a^{5}-1045a^{3}+38a-1$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}-a^{22}+14674a^{21}+22a^{20}-51359a^{19}-209a^{18}+127281a^{17}+1122a^{16}-224807a^{15}-3740a^{14}+280995a^{13}+8008a^{12}-243452a^{11}-11011a^{10}+140723a^{9}+9437a^{8}-50822a^{7}-4711a^{6}+10178a^{5}+1190a^{4}-875a^{3}-105a^{2}+14a$
| 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: | \( 3985748844865106400 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{33}\cdot(2\pi)^{0}\cdot 3985748844865106400 \cdot 1}{2\cdot\sqrt{27189028279553414235049966267283185807800188603627566700161}}\cr\approx \mathstrut & 0.103818069031133 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^33 - x^32 - 32*x^31 + 31*x^30 + 465*x^29 - 435*x^28 - 4060*x^27 + 3654*x^26 + 23751*x^25 - 20475*x^24 - 98280*x^23 + 80730*x^22 + 296010*x^21 - 230230*x^20 - 657800*x^19 + 480700*x^18 + 1081575*x^17 - 735471*x^16 - 1307504*x^15 + 817190*x^14 + 1144066*x^13 - 646646*x^12 - 705432*x^11 + 352716*x^10 + 293930*x^9 - 125970*x^8 - 77520*x^7 + 27132*x^6 + 11628*x^5 - 3060*x^4 - 816*x^3 + 136*x^2 + 17*x - 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^33 - x^32 - 32*x^31 + 31*x^30 + 465*x^29 - 435*x^28 - 4060*x^27 + 3654*x^26 + 23751*x^25 - 20475*x^24 - 98280*x^23 + 80730*x^22 + 296010*x^21 - 230230*x^20 - 657800*x^19 + 480700*x^18 + 1081575*x^17 - 735471*x^16 - 1307504*x^15 + 817190*x^14 + 1144066*x^13 - 646646*x^12 - 705432*x^11 + 352716*x^10 + 293930*x^9 - 125970*x^8 - 77520*x^7 + 27132*x^6 + 11628*x^5 - 3060*x^4 - 816*x^3 + 136*x^2 + 17*x - 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^33 - x^32 - 32*x^31 + 31*x^30 + 465*x^29 - 435*x^28 - 4060*x^27 + 3654*x^26 + 23751*x^25 - 20475*x^24 - 98280*x^23 + 80730*x^22 + 296010*x^21 - 230230*x^20 - 657800*x^19 + 480700*x^18 + 1081575*x^17 - 735471*x^16 - 1307504*x^15 + 817190*x^14 + 1144066*x^13 - 646646*x^12 - 705432*x^11 + 352716*x^10 + 293930*x^9 - 125970*x^8 - 77520*x^7 + 27132*x^6 + 11628*x^5 - 3060*x^4 - 816*x^3 + 136*x^2 + 17*x - 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^33 - x^32 - 32*x^31 + 31*x^30 + 465*x^29 - 435*x^28 - 4060*x^27 + 3654*x^26 + 23751*x^25 - 20475*x^24 - 98280*x^23 + 80730*x^22 + 296010*x^21 - 230230*x^20 - 657800*x^19 + 480700*x^18 + 1081575*x^17 - 735471*x^16 - 1307504*x^15 + 817190*x^14 + 1144066*x^13 - 646646*x^12 - 705432*x^11 + 352716*x^10 + 293930*x^9 - 125970*x^8 - 77520*x^7 + 27132*x^6 + 11628*x^5 - 3060*x^4 - 816*x^3 + 136*x^2 + 17*x - 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
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 33 |
| The 33 conjugacy class representatives for $C_{33}$ |
| Character table for $C_{33}$ is not computed |
Intermediate fields
| 3.3.4489.1, 11.11.1822837804551761449.1 |
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 | $33$ | ${\href{/padicField/3.11.0.1}{11} }^{3}$ | ${\href{/padicField/5.11.0.1}{11} }^{3}$ | $33$ | $33$ | $33$ | $33$ | $33$ | $33$ | ${\href{/padicField/29.3.0.1}{3} }^{11}$ | $33$ | ${\href{/padicField/37.3.0.1}{3} }^{11}$ | $33$ | ${\href{/padicField/43.11.0.1}{11} }^{3}$ | $33$ | ${\href{/padicField/53.11.0.1}{11} }^{3}$ | ${\href{/padicField/59.11.0.1}{11} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(67\)
| Deg $33$ | $33$ | $1$ | $32$ |