Properties

Label 73997j3
Conductor 7399773997
Discriminant 1.025×1020-1.025\times 10^{20}
j-invariant 9463555063808115539436859 \frac{9463555063808}{115539436859}
CM no
Rank 00
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+y=x3x2+423481x+475373127y^2+y=x^3-x^2+423481x+475373127 Copy content Toggle raw display (homogenize, simplify)
y2z+yz2=x3x2z+423481xz2+475373127z3y^2z+yz^2=x^3-x^2z+423481xz^2+475373127z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3+548830944x+22185594598032y^2=x^3+548830944x+22185594598032 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, -1, 1, 423481, 475373127])
 
gp: E = ellinit([0, -1, 1, 423481, 475373127])
 
magma: E := EllipticCurve([0, -1, 1, 423481, 475373127]);
 
oscar: E = elliptic_curve([0, -1, 1, 423481, 475373127])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

trivial

magma: MordellWeilGroup(E);
 

Invariants

Conductor: NN  =  73997 73997  = 7113127 \cdot 11 \cdot 31^{2}
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  102541675513029577979-102541675513029577979 = 172119316-1 \cdot 7^{2} \cdot 11^{9} \cdot 31^{6}
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  9463555063808115539436859 \frac{9463555063808}{115539436859}  = 2157211966132^{15} \cdot 7^{-2} \cdot 11^{-9} \cdot 661^{3}
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}})  =  Z\Z    (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 2.52060622152527867788474634602.5206062215252786778847463460
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 0.803612619282705554920164183730.80361261928270555492016418373
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 1.06592658831543961.0659265883154396
Szpiro ratio: σm\sigma_{m} ≈ 4.7702471256385994.770247125638599

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 0 0
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 0 0
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) = 11
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 0.139455340090153801516303473490.13945534009015380151630347349
comment: Real Period
 
sage: E.period_lattice().omega()
 
gp: if(E.disc>0,2,1)*E.omega[1]
 
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 18 18  = 2321 2\cdot3^{2}\cdot1
comment: Tamagawa numbers
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
oscar: tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 11
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Special value: L(E,1) L(E,1) ≈ 2.51019612162276842729346252292.5101961216227684272934625229
comment: Special L-value
 
r = E.rank();
 
E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  =  11    (exact)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

2.510196122L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.1394551.00000018122.510196122\displaystyle 2.510196122 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.139455 \cdot 1.000000 \cdot 18}{1^2} \approx 2.510196122

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
 
E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
 
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
 
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
 
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
 
E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
 
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
 
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
 
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form 73997.2.a.c

qq32q4+3q5+q72q9+q11+2q12+4q133q15+4q16+6q17+2q19+O(q20) q - q^{3} - 2 q^{4} + 3 q^{5} + q^{7} - 2 q^{9} + q^{11} + 2 q^{12} + 4 q^{13} - 3 q^{15} + 4 q^{16} + 6 q^{17} + 2 q^{19} + O(q^{20}) Copy content Toggle raw display

comment: q-expansion of modular form
 
sage: E.q_eigenform(20)
 
\\ actual modular form, use for small N
 
[mf,F] = mffromell(E)
 
Ser(mfcoefs(mf,20),q)
 
\\ or just the series
 
Ser(ellan(E,20),q)*q
 
magma: ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 1798200
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: no
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 3 primes pp of bad reduction:

pp Tamagawa number Kodaira symbol Reduction type Root number ordp(N)\mathrm{ord}_p(N) ordp(Δ)\mathrm{ord}_p(\Delta) ordp(den(j))\mathrm{ord}_p(\mathrm{den}(j))
77 22 I2I_{2} split multiplicative -1 1 2 2
1111 99 I9I_{9} split multiplicative -1 1 9 9
3131 11 I0I_0^{*} additive -1 2 6 0

comment: Local data
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

Galois representations

The \ell-adic Galois representation has maximal image for all primes \ell except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
33 3B 9.12.0.1

comment: mod p Galois image
 
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

gens = [[1, 18, 0, 1], [3907, 2790, 36549, 25111], [42949, 18, 42948, 19], [1, 18, 10, 181], [10, 9, 81, 73], [1, 2790, 0, 33419], [1, 0, 18, 1], [15253, 2790, 9486, 14323], [19403, 0, 0, 42965]]
 
GL(2,Integers(42966)).subgroup(gens)
 
Gens := [[1, 18, 0, 1], [3907, 2790, 36549, 25111], [42949, 18, 42948, 19], [1, 18, 10, 181], [10, 9, 81, 73], [1, 2790, 0, 33419], [1, 0, 18, 1], [15253, 2790, 9486, 14323], [19403, 0, 0, 42965]];
 
sub<GL(2,Integers(42966))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 42966=23271131 42966 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \cdot 31 , index 144144, genus 33, and generators

(11801),(390727903654925111),(42949184294819),(11810181),(1098173),(12790033419),(10181),(152532790948614323),(194030042965)\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3907 & 2790 \\ 36549 & 25111 \end{array}\right),\left(\begin{array}{rr} 42949 & 18 \\ 42948 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 2790 \\ 0 & 33419 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 15253 & 2790 \\ 9486 & 14323 \end{array}\right),\left(\begin{array}{rr} 19403 & 0 \\ 0 & 42965 \end{array}\right).

Input positive integer mm to see the generators of the reduction of HH to GL2(Z/mZ)\mathrm{GL}_2(\Z/m\Z):

The torsion field K:=Q(E[42966])K:=\Q(E[42966]) is a degree-38488736563200003848873656320000 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/42966Z)\GL_2(\Z/42966\Z).

The table below list all primes \ell for which the Serre invariants associated to the mod-\ell Galois representation are exceptional.

\ell Reduction type Serre weight Serre conductor
22 good 22 10571=11312 10571 = 11 \cdot 31^{2}
33 good 22 6727=7312 6727 = 7 \cdot 31^{2}
77 split multiplicative 88 10571=11312 10571 = 11 \cdot 31^{2}
1111 split multiplicative 1212 6727=7312 6727 = 7 \cdot 31^{2}
3131 additive 482482 77=711 77 = 7 \cdot 11

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 3 and 9.
Its isogeny class 73997j consists of 3 curves linked by isogenies of degrees dividing 9.

Twists

The minimal quadratic twist of this elliptic curve is 77b2, its twist by 31-31.

Growth of torsion in number fields

The number fields KK of degree less than 24 such that E(K)torsE(K)_{\rm tors} is strictly larger than E(Q)torsE(\Q)_{\rm tors} (which is trivial) are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
22 Q(93)\Q(\sqrt{93}) Z/3Z\Z/3\Z not in database
33 3.1.44.1 Z/2Z\Z/2\Z not in database
66 6.0.21296.1 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
66 6.0.52144051239.2 Z/3Z\Z/3\Z not in database
66 6.6.1407889383453.4 Z/9Z\Z/9\Z not in database
66 6.2.1557235152.1 Z/6Z\Z/6\Z not in database
1212 deg 12 Z/4Z\Z/4\Z not in database
1212 deg 12 Z/3ZZ/3Z\Z/3\Z \oplus \Z/3\Z not in database
1212 deg 12 Z/9Z\Z/9\Z not in database
1212 deg 12 Z/2ZZ/6Z\Z/2\Z \oplus \Z/6\Z not in database
1818 18.0.1028798609386877559094087199926788376817664.1 Z/6Z\Z/6\Z not in database
1818 18.6.20249843028561910995648918356158975620902080512.1 Z/18Z\Z/18\Z not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

pp 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss ord ord split split ord ord ord ord ord add ord ord ord ss
λ\lambda-invariant(s) 2,3 4 0 1 1 0 0 0 0 0 - 0 0 0 0,0
μ\mu-invariant(s) 0,0 0 0 0 0 0 0 0 0 0 - 0 0 0 0,0

An entry - indicates that the invariants are not computed because the reduction is additive.

pp-adic regulators

All pp-adic regulators are identically 11 since the rank is 00.