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Properties

Label 36.0.147...093.1
Degree $36$
Signature $[0, 18]$
Discriminant $1.477\times 10^{75}$
Root discriminant \(122.47\)
Ramified primes $7,37$
Class number not computed
Class group not computed
Galois group $C_{36}$ (as 36T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + x^34 - 75*x^33 + 75*x^32 - 75*x^31 + 2258*x^30 - 2258*x^29 + 2258*x^28 - 35632*x^27 + 45400*x^26 + 119990*x^25 + 198321*x^24 - 518223*x^23 - 4625406*x^22 + 3695300*x^21 + 33634*x^20 + 56706162*x^19 - 46277342*x^18 + 29299115*x^17 - 291942394*x^16 - 435767982*x^15 + 2609795075*x^14 - 3628140858*x^13 + 8655990864*x^12 - 16280549821*x^11 + 5568096849*x^10 - 10318189483*x^9 + 6830622430*x^8 + 20289170186*x^7 + 8240704677*x^6 + 9043770139*x^5 + 26040564999*x^4 + 31837565120*x^3 + 39074748065*x^2 + 19960512173*x + 3220122359)
 
gp: K = bnfinit(y^36 - y^35 + y^34 - 75*y^33 + 75*y^32 - 75*y^31 + 2258*y^30 - 2258*y^29 + 2258*y^28 - 35632*y^27 + 45400*y^26 + 119990*y^25 + 198321*y^24 - 518223*y^23 - 4625406*y^22 + 3695300*y^21 + 33634*y^20 + 56706162*y^19 - 46277342*y^18 + 29299115*y^17 - 291942394*y^16 - 435767982*y^15 + 2609795075*y^14 - 3628140858*y^13 + 8655990864*y^12 - 16280549821*y^11 + 5568096849*y^10 - 10318189483*y^9 + 6830622430*y^8 + 20289170186*y^7 + 8240704677*y^6 + 9043770139*y^5 + 26040564999*y^4 + 31837565120*y^3 + 39074748065*y^2 + 19960512173*y + 3220122359, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - x^35 + x^34 - 75*x^33 + 75*x^32 - 75*x^31 + 2258*x^30 - 2258*x^29 + 2258*x^28 - 35632*x^27 + 45400*x^26 + 119990*x^25 + 198321*x^24 - 518223*x^23 - 4625406*x^22 + 3695300*x^21 + 33634*x^20 + 56706162*x^19 - 46277342*x^18 + 29299115*x^17 - 291942394*x^16 - 435767982*x^15 + 2609795075*x^14 - 3628140858*x^13 + 8655990864*x^12 - 16280549821*x^11 + 5568096849*x^10 - 10318189483*x^9 + 6830622430*x^8 + 20289170186*x^7 + 8240704677*x^6 + 9043770139*x^5 + 26040564999*x^4 + 31837565120*x^3 + 39074748065*x^2 + 19960512173*x + 3220122359);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - x^35 + x^34 - 75*x^33 + 75*x^32 - 75*x^31 + 2258*x^30 - 2258*x^29 + 2258*x^28 - 35632*x^27 + 45400*x^26 + 119990*x^25 + 198321*x^24 - 518223*x^23 - 4625406*x^22 + 3695300*x^21 + 33634*x^20 + 56706162*x^19 - 46277342*x^18 + 29299115*x^17 - 291942394*x^16 - 435767982*x^15 + 2609795075*x^14 - 3628140858*x^13 + 8655990864*x^12 - 16280549821*x^11 + 5568096849*x^10 - 10318189483*x^9 + 6830622430*x^8 + 20289170186*x^7 + 8240704677*x^6 + 9043770139*x^5 + 26040564999*x^4 + 31837565120*x^3 + 39074748065*x^2 + 19960512173*x + 3220122359)
 

\( x^{36} - x^{35} + x^{34} - 75 x^{33} + 75 x^{32} - 75 x^{31} + 2258 x^{30} - 2258 x^{29} + \cdots + 3220122359 \) Copy content Toggle raw display

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

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 18]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1477111579412686758626382717553525114441635757641876379777829265275342440093\) \(\medspace = 7^{24}\cdot 37^{35}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(122.47\)
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:  $7^{2/3}37^{35/36}\approx 122.47269044602223$
Ramified primes:   \(7\), \(37\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{37}) \)
$\card{ \Gal(K/\Q) }$:  $36$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(259=7\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{259}(256,·)$, $\chi_{259}(1,·)$, $\chi_{259}(2,·)$, $\chi_{259}(4,·)$, $\chi_{259}(134,·)$, $\chi_{259}(8,·)$, $\chi_{259}(9,·)$, $\chi_{259}(128,·)$, $\chi_{259}(130,·)$, $\chi_{259}(16,·)$, $\chi_{259}(18,·)$, $\chi_{259}(151,·)$, $\chi_{259}(29,·)$, $\chi_{259}(32,·)$, $\chi_{259}(162,·)$, $\chi_{259}(163,·)$, $\chi_{259}(36,·)$, $\chi_{259}(170,·)$, $\chi_{259}(43,·)$, $\chi_{259}(172,·)$, $\chi_{259}(58,·)$, $\chi_{259}(64,·)$, $\chi_{259}(65,·)$, $\chi_{259}(67,·)$, $\chi_{259}(72,·)$, $\chi_{259}(205,·)$, $\chi_{259}(81,·)$, $\chi_{259}(211,·)$, $\chi_{259}(85,·)$, $\chi_{259}(86,·)$, $\chi_{259}(144,·)$, $\chi_{259}(232,·)$, $\chi_{259}(235,·)$, $\chi_{259}(116,·)$, $\chi_{259}(247,·)$, $\chi_{259}(253,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{21535333}a^{33}+\frac{8420221}{21535333}a^{32}-\frac{9957264}{21535333}a^{31}+\frac{7448375}{21535333}a^{30}+\frac{5143441}{21535333}a^{29}+\frac{7585013}{21535333}a^{28}-\frac{8469795}{21535333}a^{27}+\frac{9200200}{21535333}a^{26}+\frac{7564718}{21535333}a^{25}+\frac{7542327}{21535333}a^{24}-\frac{5933350}{21535333}a^{23}+\frac{2511162}{21535333}a^{22}+\frac{2841293}{21535333}a^{21}+\frac{6601518}{21535333}a^{20}+\frac{6071336}{21535333}a^{19}-\frac{10017510}{21535333}a^{18}-\frac{1613406}{21535333}a^{17}+\frac{9487978}{21535333}a^{16}+\frac{8925431}{21535333}a^{15}+\frac{124518}{21535333}a^{14}+\frac{7900311}{21535333}a^{13}-\frac{4892444}{21535333}a^{12}+\frac{6166228}{21535333}a^{11}+\frac{7347501}{21535333}a^{10}-\frac{10460782}{21535333}a^{9}+\frac{1006539}{21535333}a^{8}-\frac{2360898}{21535333}a^{7}-\frac{9808575}{21535333}a^{6}+\frac{4058399}{21535333}a^{5}-\frac{1207053}{21535333}a^{4}+\frac{116921}{21535333}a^{3}+\frac{9231543}{21535333}a^{2}+\frac{7318073}{21535333}a-\frac{2707890}{21535333}$, $\frac{1}{294840244103}a^{34}-\frac{3163}{294840244103}a^{33}-\frac{119451314448}{294840244103}a^{32}+\frac{107256414994}{294840244103}a^{31}-\frac{87321118666}{294840244103}a^{30}+\frac{141099027879}{294840244103}a^{29}+\frac{83700411977}{294840244103}a^{28}-\frac{8932366416}{294840244103}a^{27}-\frac{133624208210}{294840244103}a^{26}+\frac{106828941334}{294840244103}a^{25}+\frac{127426324068}{294840244103}a^{24}-\frac{98923148645}{294840244103}a^{23}-\frac{110972691938}{294840244103}a^{22}-\frac{134211259700}{294840244103}a^{21}+\frac{147194516101}{294840244103}a^{20}-\frac{9303477976}{294840244103}a^{19}+\frac{125439779917}{294840244103}a^{18}-\frac{121371029882}{294840244103}a^{17}+\frac{139569939335}{294840244103}a^{16}+\frac{132993728917}{294840244103}a^{15}+\frac{85884737835}{294840244103}a^{14}+\frac{81239310492}{294840244103}a^{13}-\frac{8181997935}{294840244103}a^{12}-\frac{72993962879}{294840244103}a^{11}-\frac{50702782688}{294840244103}a^{10}+\frac{12608421184}{294840244103}a^{9}+\frac{123541938647}{294840244103}a^{8}-\frac{131126665231}{294840244103}a^{7}+\frac{43366580378}{294840244103}a^{6}-\frac{101676688166}{294840244103}a^{5}-\frac{38790859750}{294840244103}a^{4}-\frac{14866780135}{294840244103}a^{3}-\frac{68384873330}{294840244103}a^{2}+\frac{87939570247}{294840244103}a+\frac{98176693964}{294840244103}$, $\frac{1}{91\!\cdots\!57}a^{35}-\frac{10\!\cdots\!98}{91\!\cdots\!57}a^{34}+\frac{96\!\cdots\!58}{91\!\cdots\!57}a^{33}+\frac{31\!\cdots\!43}{91\!\cdots\!57}a^{32}-\frac{40\!\cdots\!32}{91\!\cdots\!57}a^{31}-\frac{31\!\cdots\!63}{91\!\cdots\!57}a^{30}+\frac{22\!\cdots\!74}{91\!\cdots\!57}a^{29}-\frac{19\!\cdots\!84}{91\!\cdots\!57}a^{28}+\frac{14\!\cdots\!11}{91\!\cdots\!57}a^{27}-\frac{78\!\cdots\!32}{91\!\cdots\!57}a^{26}-\frac{16\!\cdots\!34}{91\!\cdots\!57}a^{25}-\frac{40\!\cdots\!85}{91\!\cdots\!57}a^{24}+\frac{18\!\cdots\!15}{91\!\cdots\!57}a^{23}+\frac{24\!\cdots\!49}{91\!\cdots\!57}a^{22}+\frac{23\!\cdots\!14}{91\!\cdots\!57}a^{21}-\frac{57\!\cdots\!13}{91\!\cdots\!57}a^{20}-\frac{41\!\cdots\!99}{91\!\cdots\!57}a^{19}-\frac{28\!\cdots\!44}{91\!\cdots\!57}a^{18}-\frac{27\!\cdots\!73}{91\!\cdots\!57}a^{17}-\frac{41\!\cdots\!75}{91\!\cdots\!57}a^{16}+\frac{15\!\cdots\!50}{91\!\cdots\!57}a^{15}+\frac{23\!\cdots\!22}{91\!\cdots\!57}a^{14}-\frac{11\!\cdots\!47}{91\!\cdots\!57}a^{13}-\frac{35\!\cdots\!26}{91\!\cdots\!57}a^{12}-\frac{21\!\cdots\!32}{91\!\cdots\!57}a^{11}+\frac{37\!\cdots\!57}{91\!\cdots\!57}a^{10}+\frac{36\!\cdots\!86}{91\!\cdots\!57}a^{9}+\frac{31\!\cdots\!15}{91\!\cdots\!57}a^{8}+\frac{33\!\cdots\!82}{91\!\cdots\!57}a^{7}-\frac{40\!\cdots\!59}{91\!\cdots\!57}a^{6}+\frac{15\!\cdots\!01}{91\!\cdots\!57}a^{5}-\frac{35\!\cdots\!69}{91\!\cdots\!57}a^{4}-\frac{38\!\cdots\!71}{91\!\cdots\!57}a^{3}+\frac{24\!\cdots\!99}{91\!\cdots\!57}a^{2}-\frac{41\!\cdots\!53}{91\!\cdots\!57}a+\frac{30\!\cdots\!73}{91\!\cdots\!57}$ Copy content Toggle raw display

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

not computed

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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + x^34 - 75*x^33 + 75*x^32 - 75*x^31 + 2258*x^30 - 2258*x^29 + 2258*x^28 - 35632*x^27 + 45400*x^26 + 119990*x^25 + 198321*x^24 - 518223*x^23 - 4625406*x^22 + 3695300*x^21 + 33634*x^20 + 56706162*x^19 - 46277342*x^18 + 29299115*x^17 - 291942394*x^16 - 435767982*x^15 + 2609795075*x^14 - 3628140858*x^13 + 8655990864*x^12 - 16280549821*x^11 + 5568096849*x^10 - 10318189483*x^9 + 6830622430*x^8 + 20289170186*x^7 + 8240704677*x^6 + 9043770139*x^5 + 26040564999*x^4 + 31837565120*x^3 + 39074748065*x^2 + 19960512173*x + 3220122359)
 
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^36 - x^35 + x^34 - 75*x^33 + 75*x^32 - 75*x^31 + 2258*x^30 - 2258*x^29 + 2258*x^28 - 35632*x^27 + 45400*x^26 + 119990*x^25 + 198321*x^24 - 518223*x^23 - 4625406*x^22 + 3695300*x^21 + 33634*x^20 + 56706162*x^19 - 46277342*x^18 + 29299115*x^17 - 291942394*x^16 - 435767982*x^15 + 2609795075*x^14 - 3628140858*x^13 + 8655990864*x^12 - 16280549821*x^11 + 5568096849*x^10 - 10318189483*x^9 + 6830622430*x^8 + 20289170186*x^7 + 8240704677*x^6 + 9043770139*x^5 + 26040564999*x^4 + 31837565120*x^3 + 39074748065*x^2 + 19960512173*x + 3220122359, 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^36 - x^35 + x^34 - 75*x^33 + 75*x^32 - 75*x^31 + 2258*x^30 - 2258*x^29 + 2258*x^28 - 35632*x^27 + 45400*x^26 + 119990*x^25 + 198321*x^24 - 518223*x^23 - 4625406*x^22 + 3695300*x^21 + 33634*x^20 + 56706162*x^19 - 46277342*x^18 + 29299115*x^17 - 291942394*x^16 - 435767982*x^15 + 2609795075*x^14 - 3628140858*x^13 + 8655990864*x^12 - 16280549821*x^11 + 5568096849*x^10 - 10318189483*x^9 + 6830622430*x^8 + 20289170186*x^7 + 8240704677*x^6 + 9043770139*x^5 + 26040564999*x^4 + 31837565120*x^3 + 39074748065*x^2 + 19960512173*x + 3220122359);
 
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^36 - x^35 + x^34 - 75*x^33 + 75*x^32 - 75*x^31 + 2258*x^30 - 2258*x^29 + 2258*x^28 - 35632*x^27 + 45400*x^26 + 119990*x^25 + 198321*x^24 - 518223*x^23 - 4625406*x^22 + 3695300*x^21 + 33634*x^20 + 56706162*x^19 - 46277342*x^18 + 29299115*x^17 - 291942394*x^16 - 435767982*x^15 + 2609795075*x^14 - 3628140858*x^13 + 8655990864*x^12 - 16280549821*x^11 + 5568096849*x^10 - 10318189483*x^9 + 6830622430*x^8 + 20289170186*x^7 + 8240704677*x^6 + 9043770139*x^5 + 26040564999*x^4 + 31837565120*x^3 + 39074748065*x^2 + 19960512173*x + 3220122359);
 
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

$C_{36}$ (as 36T1):

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 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.1369.1, 4.0.50653.1, 6.6.69343957.1, 9.9.413239695274351729.1, 12.0.177917621779460413.1, 18.18.6318380692766245764071464704595709317.1

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 $36$ $18^{2}$ $36$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ $36$ $36$ $36$ ${\href{/padicField/23.12.0.1}{12} }^{3}$ ${\href{/padicField/29.12.0.1}{12} }^{3}$ ${\href{/padicField/31.12.0.1}{12} }^{3}$ R $18^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{9}$ ${\href{/padicField/47.3.0.1}{3} }^{12}$ ${\href{/padicField/53.9.0.1}{9} }^{4}$ $36$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display 7.9.6.2$x^{9} + 252 x^{6} - 2352 x^{3} + 5488$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
7.9.6.2$x^{9} + 252 x^{6} - 2352 x^{3} + 5488$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
7.9.6.2$x^{9} + 252 x^{6} - 2352 x^{3} + 5488$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
7.9.6.2$x^{9} + 252 x^{6} - 2352 x^{3} + 5488$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
\(37\) Copy content Toggle raw display Deg $36$$36$$1$$35$