Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 + 890*x^12 - 8236*x^10 + 41097*x^8 - 111928*x^6 + 161520*x^4 - 114256*x^2 + 30976)
gp: K = bnfinit(y^16 - 48*y^14 + 890*y^12 - 8236*y^10 + 41097*y^8 - 111928*y^6 + 161520*y^4 - 114256*y^2 + 30976, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 48*x^14 + 890*x^12 - 8236*x^10 + 41097*x^8 - 111928*x^6 + 161520*x^4 - 114256*x^2 + 30976);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 48*x^14 + 890*x^12 - 8236*x^10 + 41097*x^8 - 111928*x^6 + 161520*x^4 - 114256*x^2 + 30976)
\( x^{16} - 48x^{14} + 890x^{12} - 8236x^{10} + 41097x^{8} - 111928x^{6} + 161520x^{4} - 114256x^{2} + 30976 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(111014777759270080047206560000\)
\(\medspace = 2^{8}\cdot 5^{4}\cdot 97^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(65.36\) | 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: | $2^{2/3}5^{1/2}97^{3/4}\approx 109.71107402649531$ | ||
| Ramified primes: |
\(2\), \(5\), \(97\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}$, $\frac{1}{12}a^{8}-\frac{5}{12}a^{4}-\frac{1}{2}a^{3}-\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{12}a^{9}+\frac{1}{12}a^{5}+\frac{1}{3}a^{3}-\frac{1}{2}a^{2}+\frac{1}{3}a$, $\frac{1}{12}a^{10}+\frac{1}{12}a^{6}+\frac{1}{3}a^{4}-\frac{1}{2}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{24}a^{11}-\frac{1}{24}a^{10}-\frac{1}{24}a^{9}-\frac{1}{24}a^{8}+\frac{1}{24}a^{7}+\frac{5}{24}a^{6}-\frac{1}{8}a^{5}+\frac{1}{24}a^{4}-\frac{1}{4}a^{3}-\frac{1}{3}a^{2}-\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{24}a^{12}+\frac{1}{6}a^{6}-\frac{3}{8}a^{4}+\frac{1}{3}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{48}a^{13}-\frac{1}{24}a^{9}+\frac{1}{12}a^{7}-\frac{1}{4}a^{6}-\frac{11}{48}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{147455328}a^{14}+\frac{90028}{4607979}a^{12}-\frac{632477}{73727664}a^{10}-\frac{1}{24}a^{9}+\frac{192797}{6143972}a^{8}-\frac{1}{8}a^{7}-\frac{24062723}{147455328}a^{6}+\frac{5}{24}a^{5}+\frac{521111}{12287944}a^{4}+\frac{11}{24}a^{3}+\frac{2904061}{9215958}a^{2}+\frac{1}{3}a-\frac{260847}{1535993}$, $\frac{1}{6488034432}a^{15}+\frac{22507}{50687769}a^{13}-\frac{19666737}{1081339072}a^{11}-\frac{1}{24}a^{10}+\frac{30340649}{1622008608}a^{9}-\frac{1}{24}a^{8}+\frac{559614617}{6488034432}a^{7}+\frac{5}{24}a^{6}+\frac{42500363}{270334768}a^{5}-\frac{11}{24}a^{4}-\frac{48809759}{101375538}a^{3}+\frac{1}{6}a^{2}-\frac{18461005}{405502152}a-\frac{1}{6}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $15$ | 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: |
$\frac{45645583}{6488034432}a^{15}-\frac{107785}{73727664}a^{14}-\frac{267862559}{811004304}a^{13}+\frac{1278373}{18431916}a^{12}+\frac{6394720289}{1081339072}a^{11}-\frac{1938486}{1535993}a^{10}-\frac{83895244597}{1622008608}a^{9}+\frac{139073971}{12287944}a^{8}+\frac{1523582701943}{6488034432}a^{7}-\frac{3938667647}{73727664}a^{6}-\frac{9152929386}{16895923}a^{5}+\frac{4819867375}{36863832}a^{4}+\frac{116146726067}{202751076}a^{3}-\frac{2758488983}{18431916}a^{2}-\frac{87039938347}{405502152}a+\frac{175879301}{3071986}$, $\frac{8561273}{2162678144}a^{15}+\frac{982667}{147455328}a^{14}-\frac{24755463}{135167384}a^{13}-\frac{1904191}{6143972}a^{12}+\frac{3465412013}{1081339072}a^{11}+\frac{403145569}{73727664}a^{10}-\frac{43815193469}{1622008608}a^{9}-\frac{860969059}{18431916}a^{8}+\frac{249427008177}{2162678144}a^{7}+\frac{30078552095}{147455328}a^{6}-\frac{194052221851}{811004304}a^{5}-\frac{16232311807}{36863832}a^{4}+\frac{85326858079}{405502152}a^{3}+\frac{3828138557}{9215958}a^{2}-\frac{25789722203}{405502152}a-\frac{1256412907}{9215958}$, $\frac{7689359}{1081339072}a^{15}+\frac{1323857}{147455328}a^{14}+\frac{271129495}{811004304}a^{13}-\frac{3893671}{9215958}a^{12}-\frac{9729711613}{1622008608}a^{11}+\frac{559717781}{73727664}a^{10}+\frac{42683194585}{811004304}a^{9}-\frac{2460917479}{36863832}a^{8}-\frac{778721445061}{3244017216}a^{7}+\frac{15019850699}{49151776}a^{6}+\frac{452011056653}{811004304}a^{5}-\frac{13158796367}{18431916}a^{4}-\frac{240608559637}{405502152}a^{3}+\frac{4725351895}{6143972}a^{2}+\frac{14802219853}{67583692}a-\frac{1341319774}{4607979}$, $\frac{6469351}{6488034432}a^{15}-\frac{13765}{73727664}a^{14}-\frac{12402463}{270334768}a^{13}+\frac{366917}{36863832}a^{12}+\frac{2587292555}{3244017216}a^{11}-\frac{955298}{4607979}a^{10}-\frac{10835872901}{1622008608}a^{9}+\frac{79406045}{36863832}a^{8}+\frac{184532586511}{6488034432}a^{7}-\frac{873664435}{73727664}a^{6}-\frac{8028174213}{135167384}a^{5}+\frac{156283084}{4607979}a^{4}+\frac{1803050651}{33791846}a^{3}-\frac{277233191}{6143972}a^{2}-\frac{2071596181}{135167384}a+\frac{187958963}{9215958}$, $\frac{19317821}{6488034432}a^{15}-\frac{461225}{147455328}a^{14}-\frac{57194111}{405502152}a^{13}+\frac{5427703}{36863832}a^{12}+\frac{8306999929}{3244017216}a^{11}-\frac{65040451}{24575888}a^{10}-\frac{37120554791}{1622008608}a^{9}+\frac{858226645}{36863832}a^{8}+\frac{232430015127}{2162678144}a^{7}-\frac{15728862089}{147455328}a^{6}-\frac{211980305399}{811004304}a^{5}+\frac{9221177009}{36863832}a^{4}+\frac{121532307043}{405502152}a^{3}-\frac{1258597717}{4607979}a^{2}-\frac{50302970741}{405502152}a+\frac{1002678599}{9215958}$, $\frac{15755263}{6488034432}a^{15}-\frac{23485}{36863832}a^{14}+\frac{15508247}{135167384}a^{13}+\frac{1113097}{36863832}a^{12}-\frac{6723704011}{3244017216}a^{11}-\frac{2520479}{4607979}a^{10}+\frac{29760033845}{1622008608}a^{9}+\frac{22313359}{4607979}a^{8}-\frac{549056233271}{6488034432}a^{7}-\frac{272475741}{12287944}a^{6}+\frac{161525981137}{811004304}a^{5}+\frac{630222693}{12287944}a^{4}-\frac{21909619009}{101375538}a^{3}-\frac{508187293}{9215958}a^{2}+\frac{11336094353}{135167384}a+\frac{205912175}{9215958}$, $\frac{8994509}{6488034432}a^{15}+\frac{15389}{147455328}a^{14}-\frac{4415123}{67583692}a^{13}-\frac{24794}{4607979}a^{12}+\frac{3818820073}{3244017216}a^{11}+\frac{8079053}{73727664}a^{10}-\frac{16886619599}{1622008608}a^{9}-\frac{41912483}{36863832}a^{8}+\frac{312742564757}{6488034432}a^{7}+\frac{316087951}{49151776}a^{6}-\frac{93366636125}{811004304}a^{5}-\frac{362557759}{18431916}a^{4}+\frac{52371650123}{405502152}a^{3}+\frac{278893127}{9215958}a^{2}-\frac{21442686961}{405502152}a-\frac{51937317}{3071986}$, $\frac{4349299}{1622008608}a^{15}+\frac{154245}{49151776}a^{14}-\frac{51497147}{405502152}a^{13}-\frac{5372507}{36863832}a^{12}+\frac{1867551467}{811004304}a^{11}+\frac{189324757}{73727664}a^{10}-\frac{2078287397}{101375538}a^{9}-\frac{134793015}{6143972}a^{8}+\frac{154618800191}{1622008608}a^{7}+\frac{4737346561}{49151776}a^{6}-\frac{7654415069}{33791846}a^{5}-\frac{3933309797}{18431916}a^{4}+\frac{100231979069}{405502152}a^{3}+\frac{2007117853}{9215958}a^{2}-\frac{4872981386}{50687769}a-\frac{370470191}{4607979}$, $\frac{4349299}{1622008608}a^{15}-\frac{154245}{49151776}a^{14}-\frac{51497147}{405502152}a^{13}+\frac{5372507}{36863832}a^{12}+\frac{1867551467}{811004304}a^{11}-\frac{189324757}{73727664}a^{10}-\frac{2078287397}{101375538}a^{9}+\frac{134793015}{6143972}a^{8}+\frac{154618800191}{1622008608}a^{7}-\frac{4737346561}{49151776}a^{6}-\frac{7654415069}{33791846}a^{5}+\frac{3933309797}{18431916}a^{4}+\frac{100231979069}{405502152}a^{3}-\frac{2007117853}{9215958}a^{2}-\frac{4872981386}{50687769}a+\frac{370470191}{4607979}$, $\frac{2011333}{270334768}a^{15}+\frac{207191}{147455328}a^{14}-\frac{17708648}{50687769}a^{13}-\frac{2397145}{36863832}a^{12}+\frac{423080423}{67583692}a^{11}+\frac{28060101}{24575888}a^{10}-\frac{5562217973}{101375538}a^{9}-\frac{119804909}{12287944}a^{8}+\frac{203117802595}{811004304}a^{7}+\frac{6391464935}{147455328}a^{6}-\frac{118723766461}{202751076}a^{5}-\frac{3721851283}{36863832}a^{4}+\frac{259021957043}{405502152}a^{3}+\frac{183166626}{1535993}a^{2}-\frac{12720026855}{50687769}a-\frac{263294033}{4607979}$, $\frac{5213767}{2162678144}a^{15}+\frac{51585}{24575888}a^{14}-\frac{91936259}{811004304}a^{13}-\frac{1223281}{12287944}a^{12}+\frac{2201410971}{1081339072}a^{11}+\frac{8335937}{4607979}a^{10}-\frac{9682796053}{540669536}a^{9}-\frac{595728785}{36863832}a^{8}+\frac{533521287821}{6488034432}a^{7}+\frac{5573522093}{73727664}a^{6}-\frac{26212196693}{135167384}a^{5}-\frac{3355267727}{18431916}a^{4}+\frac{43360810139}{202751076}a^{3}+\frac{3774863569}{18431916}a^{2}-\frac{34508649053}{405502152}a-\frac{765805931}{9215958}$, $\frac{21203635}{6488034432}a^{15}+\frac{279349}{49151776}a^{14}+\frac{31216529}{202751076}a^{13}-\frac{9926717}{36863832}a^{12}-\frac{8996571847}{3244017216}a^{11}+\frac{120192785}{24575888}a^{10}+\frac{39762398537}{1622008608}a^{9}-\frac{1612481867}{36863832}a^{8}-\frac{736214963563}{6488034432}a^{7}+\frac{30335714407}{147455328}a^{6}+\frac{220097521543}{811004304}a^{5}-\frac{18496630307}{36863832}a^{4}-\frac{124395758855}{405502152}a^{3}+\frac{1775153303}{3071986}a^{2}+\frac{51829737227}{405502152}a-\frac{747072689}{3071986}$, $\frac{101035}{294910656}a^{15}+\frac{78551}{147455328}a^{14}+\frac{85783}{4607979}a^{13}-\frac{304915}{12287944}a^{12}-\frac{59506811}{147455328}a^{11}+\frac{10749537}{24575888}a^{10}+\frac{331548481}{73727664}a^{9}-\frac{45531463}{12287944}a^{8}-\frac{2702896369}{98303552}a^{7}+\frac{773815013}{49151776}a^{6}+\frac{3332814067}{36863832}a^{5}-\frac{1165552561}{36863832}a^{4}-\frac{5270544589}{36863832}a^{3}+\frac{233902145}{9215958}a^{2}+\frac{1427590003}{18431916}a-\frac{34266070}{4607979}$, $\frac{1316511}{2162678144}a^{15}+\frac{459715}{147455328}a^{14}+\frac{2047537}{67583692}a^{13}-\frac{1827633}{12287944}a^{12}-\frac{1910479937}{3244017216}a^{11}+\frac{200917949}{73727664}a^{10}+\frac{9359861323}{1622008608}a^{9}-\frac{113461900}{4607979}a^{8}-\frac{198309890773}{6488034432}a^{7}+\frac{17245684495}{147455328}a^{6}+\frac{23399841973}{270334768}a^{5}-\frac{440655601}{1535993}a^{4}-\frac{16094497469}{135167384}a^{3}+\frac{1009091005}{3071986}a^{2}+\frac{23606973433}{405502152}a-\frac{401504363}{3071986}$, $\frac{8972313}{540669536}a^{15}+\frac{1337849}{147455328}a^{14}-\frac{634174265}{811004304}a^{13}-\frac{15850549}{36863832}a^{12}+\frac{11421118789}{811004304}a^{11}+\frac{575851193}{73727664}a^{10}-\frac{8402743127}{67583692}a^{9}-\frac{858393817}{12287944}a^{8}+\frac{929984625091}{1622008608}a^{7}+\frac{16147594027}{49151776}a^{6}-\frac{1102108221293}{811004304}a^{5}-\frac{9852819647}{12287944}a^{4}+\frac{203894755121}{135167384}a^{3}+\frac{17095367765}{18431916}a^{2}-\frac{61927212665}{101375538}a-\frac{1801337249}{4607979}$
| 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: | \( 5190457249.388809 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 5190457249.388809 \cdot 1}{2\cdot\sqrt{111014777759270080047206560000}}\cr\approx \mathstrut & 0.510464043633944 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 + 890*x^12 - 8236*x^10 + 41097*x^8 - 111928*x^6 + 161520*x^4 - 114256*x^2 + 30976)
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^16 - 48*x^14 + 890*x^12 - 8236*x^10 + 41097*x^8 - 111928*x^6 + 161520*x^4 - 114256*x^2 + 30976, 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^16 - 48*x^14 + 890*x^12 - 8236*x^10 + 41097*x^8 - 111928*x^6 + 161520*x^4 - 114256*x^2 + 30976);
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^16 - 48*x^14 + 890*x^12 - 8236*x^10 + 41097*x^8 - 111928*x^6 + 161520*x^4 - 114256*x^2 + 30976);
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
$C_4\times S_4$ (as 16T181):
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 solvable group of order 96 |
| The 20 conjugacy class representatives for $C_4\times S_4$ |
| Character table for $C_4\times S_4$ |
Intermediate fields
| \(\Q(\sqrt{97}) \), 4.4.912673.1, 4.4.18253460.1, 8.8.333188801971600.1 |
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]
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Minimal sibling: | 12.12.12927725516498080000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(5\)
| 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(97\)
| 97.16.12.1 | $x^{16} + 24 x^{14} + 320 x^{13} + 624 x^{12} + 5760 x^{11} + 37296 x^{10} - 302080 x^{9} + 509608 x^{8} + 2547200 x^{7} + 22903776 x^{6} - 8963840 x^{5} + 115312576 x^{4} + 195816960 x^{3} + 1860806080 x^{2} + 1868492800 x + 5486792336$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |