Properties

Label 111090.l1
Conductor 111090111090
Discriminant 2.620×1023-2.620\times 10^{23}
j-invariant 336029669236202135296324810240 -\frac{33602966923620213529}{6324810240}
CM no
Rank 11
Torsion structure trivial

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy=x3+x22326184142x43184119407564y^2+xy=x^3+x^2-2326184142x-43184119407564 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz=x3+x2z2326184142xz243184119407564z3y^2z+xyz=x^3+x^2z-2326184142xz^2-43184119407564z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x33014734648707x2014753054059578754y^2=x^3-3014734648707x-2014753054059578754 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 1, 0, -2326184142, -43184119407564])
 
gp: E = ellinit([1, 1, 0, -2326184142, -43184119407564])
 
magma: E := EllipticCurve([1, 1, 0, -2326184142, -43184119407564]);
 
oscar: E = elliptic_curve([1, 1, 0, -2326184142, -43184119407564])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

Z\Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(6996409804533344795784282062906976115685293541090118077792677481521744898824956341948420389027529243362355394144437308365756742128503143382509094070546902972840790401877668975/125497966090630596192190099740585100188948813213759943286340035734525663240912348233389602276999588311096369538734689531536603611246430342981635668205316837247825604104881,854203150932830263075905593409543125484607764898592251883306775348081740339765606281291117062203946576785576151884920022128714274686908936317582762572740194005992339476217699135894589116129158627297170191381920684339163227338739732164320819266006641627307678999/1405901942816196429865092410607312454628190087502355293557351544334208304608342112959247303167258874715446630895292819695383314562648186104975987863918462010779633673778709442051114949515013615715805360838607658354514701340707810140876978790353092918375529)(6996409804533344795784282062906976115685293541090118077792677481521744898824956341948420389027529243362355394144437308365756742128503143382509094070546902972840790401877668975/125497966090630596192190099740585100188948813213759943286340035734525663240912348233389602276999588311096369538734689531536603611246430342981635668205316837247825604104881, 854203150932830263075905593409543125484607764898592251883306775348081740339765606281291117062203946576785576151884920022128714274686908936317582762572740194005992339476217699135894589116129158627297170191381920684339163227338739732164320819266006641627307678999/1405901942816196429865092410607312454628190087502355293557351544334208304608342112959247303167258874715446630895292819695383314562648186104975987863918462010779633673778709442051114949515013615715805360838607658354514701340707810140876978790353092918375529)403.00033034405929621846242365403.00033034405929621846242365\infty

Integral points

None

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: NN  =  111090 111090  = 23572322 \cdot 3 \cdot 5 \cdot 7 \cdot 23^{2}
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  262014822331562022965760-262014822331562022965760 = 12935772310-1 \cdot 2^{9} \cdot 3 \cdot 5 \cdot 7^{7} \cdot 23^{10}
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  336029669236202135296324810240 -\frac{33602966923620213529}{6324810240}  = 1293151772323136132113-1 \cdot 2^{-9} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-7} \cdot 23^{2} \cdot 31^{3} \cdot 61^{3} \cdot 211^{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}} ≈ 3.88635235136313758049577337453.8863523513631375804957733745
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 1.27344050475551283815681268131.2734405047555128381568126813
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 1.0251084106517871.025108410651787
Szpiro ratio: σm\sigma_{m} ≈ 6.5687259572162326.568725957216232

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 1 1
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 1 1
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) ≈ 403.00033034405929621846242365403.00033034405929621846242365
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 0.0108694740841026933663791494670.010869474084102693366379149467
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 = 1 1
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) ≈ 4.38040164655957678418079726394.3804016465595767841807972639
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    (rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

4.380401647L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.010869403.0003301124.380401647\displaystyle 4.380401647 \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 0.010869 \cdot 403.000330 \cdot 1}{1^2} \approx 4.380401647

# 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 111090.2.a.l

qq2q3+q4+q5+q6q7q8+q9q10+4q11q122q13+q14q15+q16q182q19+O(q20) q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} - 2 q^{13} + q^{14} - q^{15} + q^{16} - q^{18} - 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: 61205760
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 5 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))
22 11 I9I_{9} nonsplit multiplicative 1 1 9 9
33 11 I1I_{1} nonsplit multiplicative 1 1 1 1
55 11 I1I_{1} split multiplicative -1 1 1 1
77 11 I7I_{7} nonsplit multiplicative 1 1 7 7
2323 11 IIII^{*} additive -1 2 10 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.

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 = [[631, 2, 631, 3], [839, 2, 838, 3], [1, 0, 2, 1], [1, 2, 0, 1], [337, 2, 337, 3], [281, 2, 281, 3], [241, 2, 241, 3], [421, 2, 421, 3], [1, 1, 839, 0]]
 
GL(2,Integers(840)).subgroup(gens)
 
Gens := [[631, 2, 631, 3], [839, 2, 838, 3], [1, 0, 2, 1], [1, 2, 0, 1], [337, 2, 337, 3], [281, 2, 281, 3], [241, 2, 241, 3], [421, 2, 421, 3], [1, 1, 839, 0]];
 
sub<GL(2,Integers(840))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 840=23357 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 , index 22, genus 00, and generators

(63126313),(83928383),(1021),(1201),(33723373),(28122813),(24122413),(42124213),(118390)\left(\begin{array}{rr} 631 & 2 \\ 631 & 3 \end{array}\right),\left(\begin{array}{rr} 839 & 2 \\ 838 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 2 \\ 337 & 3 \end{array}\right),\left(\begin{array}{rr} 281 & 2 \\ 281 & 3 \end{array}\right),\left(\begin{array}{rr} 241 & 2 \\ 241 & 3 \end{array}\right),\left(\begin{array}{rr} 421 & 2 \\ 421 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 839 & 0 \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[840])K:=\Q(E[840]) is a degree-3567255552035672555520 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/840Z)\GL_2(\Z/840\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 nonsplit multiplicative 44 55545=357232 55545 = 3 \cdot 5 \cdot 7 \cdot 23^{2}
33 nonsplit multiplicative 44 18515=57232 18515 = 5 \cdot 7 \cdot 23^{2}
55 split multiplicative 66 22218=237232 22218 = 2 \cdot 3 \cdot 7 \cdot 23^{2}
77 nonsplit multiplicative 88 15870=235232 15870 = 2 \cdot 3 \cdot 5 \cdot 23^{2}
2323 additive 112112 210=2357 210 = 2 \cdot 3 \cdot 5 \cdot 7

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 111090.l consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 111090.c1, its twist by 23-23.

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
33 3.1.444360.1 Z/2Z\Z/2\Z not in database
66 6.0.165862880064000.1 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
88 deg 8 Z/3Z\Z/3\Z not in database
1212 deg 12 Z/4Z\Z/4\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 nonsplit nonsplit split nonsplit ord ord ss ord add ord ord ord ord ord ss
λ\lambda-invariant(s) 2 1 2 1 1 1 1,1 1 - 3 3 1 1 1 1,1
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

Note: pp-adic regulator data only exists for primes p5p\ge 5 of good ordinary reduction.