Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2+267244x+118914005\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z+267244xz^2+118914005z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+346348197x+5542856602614\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(1069, 39809\right)\) |
$\hat{h}(P)$ | ≈ | $1.6222077237204861135145996199$ |
Integral points
\( \left(1069, 39809\right) \), \( \left(1069, -40879\right) \)
Invariants
Conductor: | \( 433698 \) | = | $2 \cdot 3 \cdot 41^{2} \cdot 43$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-7318807395170402304 $ | = | $-1 \cdot 2^{14} \cdot 3^{7} \cdot 41^{6} \cdot 43 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{444369620591}{1540767744} \) | = | $2^{-14} \cdot 3^{-7} \cdot 13^{3} \cdot 43^{-1} \cdot 587^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.3027529564830366208290193256\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $0.44596692313088271889563763908\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.6222077237204861135145996199\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.16680671868584795624849890486\dots$ | 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: | $ 28 $ = $ ( 2 \cdot 7 )\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)
|
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);
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Special value: | $ L'(E,1) $ ≈ $ 7.5766641277782809167567814691 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 7.576664128 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.166807 \cdot 1.622208 \cdot 28}{1^2} \approx 7.576664128$
Modular invariants
Modular form 433698.2.a.q
For more coefficients, see the Downloads section to the right.
Modular degree: | 11854080 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $14$ | $I_{14}$ | Split multiplicative | -1 | 1 | 14 | 14 |
$3$ | $1$ | $I_{7}$ | Non-split multiplicative | 1 | 1 | 7 | 7 |
$41$ | $2$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
$43$ | $1$ | $I_{1}$ | 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 |
---|---|---|
$7$ | 7B.6.1 | 7.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 148092 = 2^{2} \cdot 3 \cdot 7 \cdot 41 \cdot 43 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 74047 & 61418 \\ 0 & 95203 \end{array}\right),\left(\begin{array}{rr} 104756 & 104755 \\ 117383 & 43338 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6028 & 104755 \\ 92701 & 43338 \end{array}\right),\left(\begin{array}{rr} 118532 & 104755 \\ 105329 & 43338 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 148079 & 14 \\ 148078 & 15 \end{array}\right),\left(\begin{array}{rr} 32507 & 0 \\ 0 & 148091 \end{array}\right)$.
The torsion field $K:=\Q(E[148092])$ is a degree-$889825009257676800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/148092\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 433698q
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 258f1, its twist by $41$.
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.