Properties

Label 429429ct1
Conductor $429429$
Discriminant $-5.009\times 10^{16}$
j-invariant \( -\frac{30515071121161}{85766121} \)
CM no
Rank $2$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3+x^2-544183x+154661434\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3+x^2z-544183xz^2+154661434z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-705261843x+7226462788974\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 1, 0, -544183, 154661434])
 
gp: E = ellinit([1, 1, 0, -544183, 154661434])
 
magma: E := EllipticCurve([1, 1, 0, -544183, 154661434]);
 
oscar: E = EllipticCurve([1, 1, 0, -544183, 154661434])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z \oplus \Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generators and heights

$P$ =  \(\left(\frac{523}{4}, \frac{73565}{8}\right)\) Copy content Toggle raw display \(\left(394, 1072\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $3.0878997609739845139315109770$$3.8333570601232943499983622092$

sage: E.gens()
 
magma: Generators(E);
 
gp: E.gen
 

Integral points

\( \left(394, 1072\right) \), \( \left(394, -1466\right) \), \( \left(1074, 27932\right) \), \( \left(1074, -29006\right) \) Copy content Toggle raw display

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

Invariants

Conductor: \( 429429 \)  =  $3 \cdot 7 \cdot 11^{2} \cdot 13^{2}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-50091178853284569 $  =  $-1 \cdot 3^{6} \cdot 7^{6} \cdot 11^{2} \cdot 13^{6} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( -\frac{30515071121161}{85766121} \)  =  $-1 \cdot 3^{-6} \cdot 7^{-6} \cdot 11 \cdot 14051^{3}$
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.0766857093957343722415003511\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $0.39456181853190424687109936733\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 

BSD invariants

Analytic rank: $2$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $9.6485123480048915094204613401\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.35758854886837640470664599120\dots$
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: $ 4 $  = $ 2\cdot2\cdot1\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: $1$
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Analytic order of Ш: $1$ ( rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L^{(2)}(E,1)/2! $ ≈ $ 13.800790117046721261117431709 $
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);
 

BSD formula

$\displaystyle 13.800790117 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.357589 \cdot 9.648512 \cdot 4}{1^2} \approx 13.800790117$

# 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 analyiic 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 429429.2.a.ct

\( q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - q^{7} - 3 q^{8} + q^{9} - q^{10} + q^{12} - q^{14} + q^{15} - q^{16} - 7 q^{17} + q^{18} + 6 q^{19} + O(q^{20}) \) 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: 5318784
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$7$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$11$ $1$ $II$ Additive -1 2 2 0
$13$ $1$ $I_0^{*}$ 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
$2$ 2G 4.2.0.1

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, 2, 2, 5], [11087, 0, 0, 24023], [1, 4, 0, 1], [10167, 20332, 16640, 14795], [8451, 20332, 20072, 14795], [11259, 20332, 3536, 14795], [24021, 4, 24020, 5], [12013, 17329, 0, 18019], [1, 0, 4, 1], [22179, 20332, 8632, 14795], [2, 3, 24019, 24017]]
 
GL(2,Integers(24024)).subgroup(gens)
 
Gens := [[1, 2, 2, 5], [11087, 0, 0, 24023], [1, 4, 0, 1], [10167, 20332, 16640, 14795], [8451, 20332, 20072, 14795], [11259, 20332, 3536, 14795], [24021, 4, 24020, 5], [12013, 17329, 0, 18019], [1, 0, 4, 1], [22179, 20332, 8632, 14795], [2, 3, 24019, 24017]];
 
sub<GL(2,Integers(24024))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 24024 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $4$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 11087 & 0 \\ 0 & 24023 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10167 & 20332 \\ 16640 & 14795 \end{array}\right),\left(\begin{array}{rr} 8451 & 20332 \\ 20072 & 14795 \end{array}\right),\left(\begin{array}{rr} 11259 & 20332 \\ 3536 & 14795 \end{array}\right),\left(\begin{array}{rr} 24021 & 4 \\ 24020 & 5 \end{array}\right),\left(\begin{array}{rr} 12013 & 17329 \\ 0 & 18019 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 22179 & 20332 \\ 8632 & 14795 \end{array}\right),\left(\begin{array}{rr} 2 & 3 \\ 24019 & 24017 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[24024])$ is a degree-$12854962107187200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24024\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 429429ct consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 2541g1, its twist by $13$.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.