Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+6239508x+2108111024\)
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(homogenize, simplify) |
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\(y^2z=x^3+6239508xz^2+2108111024z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+6239508x+2108111024\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(358, 66240\right)\)
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| $\hat{h}(P)$ | ≈ | $0.83819518399111615357577178176$ |
Torsion generators
\( \left(-332, 0\right) \)
Integral points
\( \left(-332, 0\right) \), \((358,\pm 66240)\), \((1784,\pm 137540)\), \((2773,\pm 201825)\), \((9190,\pm 914112)\)
Invariants
| Conductor: | \( 463680 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-17466307087226476953600 $ | = | $-1 \cdot 2^{21} \cdot 3^{8} \cdot 5^{2} \cdot 7^{3} \cdot 23^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{140574743422291079}{91397357868600} \) | = | $2^{-3} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-3} \cdot 11^{3} \cdot 23^{-6} \cdot 47269^{3}$ | 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: | $2.9567568747426276776440580056\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $1.3677299595686548678205872050\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $0.9816656371849553\dots$ | |||
| Szpiro ratio: | $4.487856470953506\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $0.83819518399111615357577178176\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.076893292886278730652136384355\dots$ | 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: | $ 576 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot3\cdot( 2 \cdot 3 ) $ | 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) $ ≈ $ 9.2810286401035944253246554651 $ | 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
$\displaystyle 9.281028640 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.076893 \cdot 0.838195 \cdot 576}{2^2} \approx 9.281028640$
Modular invariants
Modular form 463680.2.a.ml
For more coefficients, see the Downloads section to the right.
| Modular degree: | 26542080 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{11}^{*}$ | Additive | 1 | 6 | 21 | 3 |
| $3$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
| $5$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
| $23$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 19320 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 19270 & 19311 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 6439 & 19308 \\ 16094 & 19247 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 15457 & 12 \\ 15462 & 73 \end{array}\right),\left(\begin{array}{rr} 19309 & 12 \\ 19308 & 13 \end{array}\right),\left(\begin{array}{rr} 4035 & 808 \\ 3998 & 797 \end{array}\right),\left(\begin{array}{rr} 8290 & 3 \\ 2733 & 19312 \end{array}\right),\left(\begin{array}{rr} 6721 & 12 \\ 1686 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19310 & 19317 \\ 9687 & 8 \end{array}\right)$.
The torsion field $K:=\Q(E[19320])$ is a degree-$198553444024320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19320\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 463680ml
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 4830o4, its twist by $-24$.
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.