Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-27074850x+155346706196\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-27074850xz^2+155346706196z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-433197603x+9941755998942\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-\frac{28061}{4}, \frac{28061}{8}\right) \)
Integral points
None
Invariants
| Conductor: | \( 422730 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 61$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-9154156647118293115240320 $ | = | $-1 \cdot 2^{7} \cdot 3^{6} \cdot 5 \cdot 7 \cdot 11^{16} \cdot 61 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{3010886403566634803117601}{12557142177117000158080} \) | = | $-1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{-16} \cdot 61^{-1} \cdot 48133067^{3}$ | 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: | $3.4748469011179621582379826760\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $2.9255407567839073125403600575\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.0144868190142529\dots$ | |||
| Szpiro ratio: | $5.022267505246248\dots$ | |||
BSD invariants
| Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.063635546189962507377375577610\dots$ | 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: | $ 4 $ = $ 1\cdot2\cdot1\cdot1\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: | $2$ | 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: | $16$ = $4^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L(E,1) $ ≈ $ 1.0181687390394001180380092418 $ | 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
$\displaystyle 1.018168739 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{16 \cdot 0.063636 \cdot 1.000000 \cdot 4}{2^2} \approx 1.018168739$
Modular invariants
Modular form 422730.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 142213120 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 6 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{7}$ | Non-split multiplicative | 1 | 1 | 7 | 7 |
| $3$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
| $5$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $2$ | $I_{16}$ | Non-split multiplicative | 1 | 1 | 16 | 16 |
| $61$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 51240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 61 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 23488 & 49113 \\ 27747 & 10654 \end{array}\right),\left(\begin{array}{rr} 34159 & 0 \\ 0 & 51239 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 21964 & 34161 \\ 2463 & 17086 \end{array}\right),\left(\begin{array}{rr} 44416 & 3 \\ 17085 & 34162 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 51234 & 51235 \end{array}\right),\left(\begin{array}{rr} 27763 & 27756 \\ 10722 & 36301 \end{array}\right),\left(\begin{array}{rr} 19048 & 3 \\ 20445 & 34162 \end{array}\right),\left(\begin{array}{rr} 51233 & 8 \\ 51232 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[51240])$ is a degree-$20237040746496000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/51240\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 422730s
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 46970p3, its twist by $-3$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.