Properties

Label 24843d4
Conductor 2484324843
Discriminant 1.227×10161.227\times 10^{16}
j-invariant 1302764097721609 \frac{13027640977}{21609}
CM no
Rank 00
Torsion structure Z/2ZZ/2Z\Z/{2}\Z \oplus \Z/{2}\Z

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy=x3+x2405941x+99238560y^2+xy=x^3+x^2-405941x+99238560 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz=x3+x2z405941xz2+99238560z3y^2z+xyz=x^3+x^2z-405941xz^2+99238560z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3526100211x+4637965755150y^2=x^3-526100211x+4637965755150 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 1, 0, -405941, 99238560])
 
gp: E = ellinit([1, 1, 0, -405941, 99238560])
 
magma: E := EllipticCurve([1, 1, 0, -405941, 99238560]);
 
oscar: E = elliptic_curve([1, 1, 0, -405941, 99238560])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

Z/2ZZ/2Z\Z/{2}\Z \oplus \Z/{2}\Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(736,368)(-736, 368)0022
(356,178)(356, -178)0022

Integral points

(736,368) \left(-736, 368\right) , (356,178) \left(356, -178\right) Copy content Toggle raw display

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

Invariants

Conductor: NN  =  24843 24843  = 3721323 \cdot 7^{2} \cdot 13^{2}
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  1227108666735396912271086667353969 = 327101363^{2} \cdot 7^{10} \cdot 13^{6}
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  1302764097721609 \frac{13027640977}{21609}  = 327413318133^{-2} \cdot 7^{-4} \cdot 13^{3} \cdot 181^{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}} ≈ 1.98306154636939213662906988641.9830615463693921366290698864
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 0.27236820688903288395035020610-0.27236820688903288395035020610
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 1.08149027405681971.0814902740568197
Szpiro ratio: σm\sigma_{m} ≈ 4.9756766556461984.975676655646198

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 ≈ 0.400652322058537142353402107550.40065232205853714235340210755
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 = 32 32  = 22222 2\cdot2^{2}\cdot2^{2}
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}} = 44
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) ≈ 0.801304644117074284706804215100.80130464411707428470680421510
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

0.801304644L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.4006521.00000032420.801304644\displaystyle 0.801304644 \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 0.400652 \cdot 1.000000 \cdot 32}{4^2} \approx 0.801304644

# 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   24843.2.a.p

q+q2q3q42q5q63q8+q92q104q11+q12+2q15q16+6q17+q18+4q19+O(q20) q + q^{2} - q^{3} - q^{4} - 2 q^{5} - q^{6} - 3 q^{8} + q^{9} - 2 q^{10} - 4 q^{11} + q^{12} + 2 q^{15} - q^{16} + 6 q^{17} + q^{18} + 4 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: 221184
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))
33 22 I2I_{2} nonsplit multiplicative 1 1 2 2
77 44 I4I_{4}^{*} additive -1 2 10 4
1313 44 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
22 2Cs 8.24.0.18

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 = [[287, 338, 1326, 1171], [2177, 8, 2176, 9], [1, 0, 8, 1], [167, 0, 0, 2183], [5, 4, 2180, 2181], [1769, 884, 2080, 261], [1015, 1560, 1170, 625], [935, 832, 884, 1143], [1, 8, 0, 1]]
 
GL(2,Integers(2184)).subgroup(gens)
 
Gens := [[287, 338, 1326, 1171], [2177, 8, 2176, 9], [1, 0, 8, 1], [167, 0, 0, 2183], [5, 4, 2180, 2181], [1769, 884, 2080, 261], [1015, 1560, 1170, 625], [935, 832, 884, 1143], [1, 8, 0, 1]];
 
sub<GL(2,Integers(2184))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 2184=233713 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 , index 192192, genus 11, and generators

(28733813261171),(2177821769),(1081),(167002183),(5421802181),(17698842080261),(101515601170625),(9358328841143),(1801)\left(\begin{array}{rr} 287 & 338 \\ 1326 & 1171 \end{array}\right),\left(\begin{array}{rr} 2177 & 8 \\ 2176 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 167 & 0 \\ 0 & 2183 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 2180 & 2181 \end{array}\right),\left(\begin{array}{rr} 1769 & 884 \\ 2080 & 261 \end{array}\right),\left(\begin{array}{rr} 1015 & 1560 \\ 1170 & 625 \end{array}\right),\left(\begin{array}{rr} 935 & 832 \\ 884 & 1143 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \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[2184])K:=\Q(E[2184]) is a degree-2028876595220288765952 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/2184Z)\GL_2(\Z/2184\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 8281=72132 8281 = 7^{2} \cdot 13^{2}
33 nonsplit multiplicative 44 8281=72132 8281 = 7^{2} \cdot 13^{2}
77 additive 3232 507=3132 507 = 3 \cdot 13^{2}
1313 additive 8686 147=372 147 = 3 \cdot 7^{2}

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2 and 4.
Its isogeny class 24843d consists of 6 curves linked by isogenies of degrees dividing 8.

Twists

The minimal quadratic twist of this elliptic curve is 21a2, its twist by 91-91.

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} Z/2ZZ/2Z\cong \Z/{2}\Z \oplus \Z/{2}\Z are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
22 Q(91)\Q(\sqrt{91}) Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
44 Q(3,91)\Q(\sqrt{3}, \sqrt{-91}) Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
44 Q(3,91)\Q(\sqrt{-3}, \sqrt{-91}) Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
88 8.0.1421970391296.10 Z/4ZZ/4Z\Z/4\Z \oplus \Z/4\Z not in database
88 8.0.364024420171776.33 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
88 8.8.4494128644096.1 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
88 8.0.12797733521664.81 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
88 deg 8 Z/2ZZ/6Z\Z/2\Z \oplus \Z/6\Z not in database
1616 deg 16 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
1616 deg 16 Z/4ZZ/8Z\Z/4\Z \oplus \Z/8\Z not in database
1616 deg 16 Z/2ZZ/12Z\Z/2\Z \oplus \Z/12\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 7 13
Reduction type ord nonsplit add add
λ\lambda-invariant(s) 5 0 - -
μ\mu-invariant(s) 0 0 - -

All Iwasawa λ\lambda and μ\mu-invariants for primes p3p\ge 3 of good reduction are zero.

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.