Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-5161401x-4729230171\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-5161401xz^2-4729230171z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6689176371x-220546625215986\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{29558479}{10201}, \frac{68796240926}{1030301}\right)\)
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| $\hat{h}(P)$ | ≈ | $13.146769961001791169505742787$ |
Integral points
None
Invariants
| Conductor: | \( 438702 \) | = | $2 \cdot 3 \cdot 11 \cdot 17^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-853156748497930956288 $ | = | $-1 \cdot 2^{9} \cdot 3^{3} \cdot 11^{3} \cdot 17^{10} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{629989223007953593}{35345595428352} \) | = | $-1 \cdot 2^{-9} \cdot 3^{-3} \cdot 11^{-3} \cdot 17^{-4} \cdot 23^{-1} \cdot 773^{3} \cdot 1109^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\Z$ | |||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $\mathrm{SU}(2)$ | |||
| Faltings height: | $2.7736453696613341370137226893\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $1.3570386976332260968889553804\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $0.9260263665058911\dots$ | |||
| Szpiro ratio: | $4.470303985477481\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $13.146769961001791169505742787\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.049918898467759642360589662946\dots$ | 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);
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| Tamagawa product: | $ 6 $ = $ 1\cdot1\cdot3\cdot2\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)
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| Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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| Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L'(E,1) $ ≈ $ 3.9376336491734448381779941949 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 3.937633649 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.049919 \cdot 13.146770 \cdot 6}{1^2} \approx 3.937633649$
Modular invariants
Modular form 438702.2.a.f
For more coefficients, see the Downloads section to the right.
| Modular degree: | 29859840 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $11$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $17$ | $2$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $23$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6072 = 2^{3} \cdot 3 \cdot 11 \cdot 23 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1519 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4049 & 2 \\ 4049 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3961 & 2 \\ 3961 & 3 \end{array}\right),\left(\begin{array}{rr} 6071 & 2 \\ 6070 & 3 \end{array}\right),\left(\begin{array}{rr} 3313 & 2 \\ 3313 & 3 \end{array}\right),\left(\begin{array}{rr} 3037 & 2 \\ 3037 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 6071 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[6072])$ is a degree-$130005231206400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6072\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 219351 = 3 \cdot 11 \cdot 17^{2} \cdot 23 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 6647 = 17^{2} \cdot 23 \) |
| $11$ | split multiplicative | $12$ | \( 39882 = 2 \cdot 3 \cdot 17^{2} \cdot 23 \) |
| $17$ | additive | $162$ | \( 1518 = 2 \cdot 3 \cdot 11 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 19074 = 2 \cdot 3 \cdot 11 \cdot 17^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 438702f consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 25806b1, its twist by $17$.
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.