Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-67298x+8250548\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-67298xz^2+8250548z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-87217587x+385199231886\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{20453}{121}, \frac{1628283}{1331}\right)\) |
$\hat{h}(P)$ | ≈ | $7.4877476741529385243569893645$ |
Torsion generators
\( \left(-307, 153\right) \)
Integral points
\( \left(-307, 153\right) \)
Invariants
Conductor: | \( 428910 \) | = | $2 \cdot 3 \cdot 5 \cdot 17 \cdot 29^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-9940496997089280 $ | = | $-1 \cdot 2^{16} \cdot 3 \cdot 5 \cdot 17 \cdot 29^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{56667352321}{16711680} \) | = | $-1 \cdot 2^{-16} \cdot 3^{-1} \cdot 5^{-1} \cdot 17^{-1} \cdot 23^{3} \cdot 167^{3}$ | 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: | $1.7840934307864830396488267824\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.10044551579324602605719076622\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0017576460079456\dots$ | |||
Szpiro ratio: | $3.498810749124448\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $7.4877476741529385243569893645\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.38642961885642591622571638116\dots$ | 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: | $ 8 $ = $ 2\cdot1\cdot1\cdot1\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)
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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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 5.7869749596320193396963246073 $ | 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
$\displaystyle 5.786974960 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.386430 \cdot 7.487748 \cdot 8}{2^2} \approx 5.786974960$
Modular invariants
Modular form 428910.2.a.bh
For more coefficients, see the Downloads section to the right.
Modular degree: | 3211264 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{16}$ | Non-split multiplicative | 1 | 1 | 16 | 16 |
$3$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$17$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$29$ | $4$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
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 | 32.48.0.8 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 236640 = 2^{5} \cdot 3 \cdot 5 \cdot 17 \cdot 29 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 27232 & 29 \\ 114347 & 124962 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 234078 & 234635 \end{array}\right),\left(\begin{array}{rr} 236609 & 32 \\ 236608 & 33 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 97441 & 40832 \\ 222314 & 33119 \end{array}\right),\left(\begin{array}{rr} 30 & 106111 \\ 60233 & 73284 \end{array}\right),\left(\begin{array}{rr} 76503 & 106082 \\ 190762 & 32655 \end{array}\right),\left(\begin{array}{rr} 171359 & 0 \\ 0 & 236639 \end{array}\right),\left(\begin{array}{rr} 160342 & 40803 \\ 143405 & 179308 \end{array}\right)$.
The torsion field $K:=\Q(E[236640])$ is a degree-$630302664189542400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/236640\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 428910bh
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 510e1, its twist by $29$.
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.