Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-3192730158x+71988485212687\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-3192730158xz^2+71988485212687z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-258611142825x+52478829886620375\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{21749221907979672827539965417}{1488627780496092790526281}, \frac{9689459763055577502694809168615103290662875}{1816264854255379862233110101742960571}\right)\)
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| $\hat{h}(P)$ | ≈ | $62.799208995059607825340352214$ |
Integral points
None
Invariants
| Conductor: | \( 450800 \) | = | $2^{4} \cdot 5^{2} \cdot 7^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-155814195500968173969143750000 $ | = | $-1 \cdot 2^{4} \cdot 5^{8} \cdot 7^{12} \cdot 23^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{122372013839654770813696}{5297595236711512175} \) | = | $-1 \cdot 2^{8} \cdot 5^{-2} \cdot 7^{-6} \cdot 23^{-9} \cdot 53^{3} \cdot 151^{3} \cdot 977^{3}$ | 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: | $4.3672031101522739762711011341\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $2.3584800192209186999456343886\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.0037356862651234\dots$ | |||
| Szpiro ratio: | $5.940508721762801\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $62.799208995059607825340352214\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.032136230082037564860224702058\dots$ | 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\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) $ ≈ $ 8.0725193169407944075517859803 $ | 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
$\displaystyle 8.072519317 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.032136 \cdot 62.799209 \cdot 4}{1^2} \approx 8.072519317$
Modular invariants
Modular form 450800.2.a.cs
For more coefficients, see the Downloads section to the right.
| Modular degree: | 489701376 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | Additive | -1 | 4 | 4 | 0 |
| $5$ | $2$ | $I_{2}^{*}$ | Additive | 1 | 2 | 8 | 2 |
| $7$ | $2$ | $I_{6}^{*}$ | Additive | -1 | 2 | 12 | 6 |
| $23$ | $1$ | $I_{9}$ | Non-split multiplicative | 1 | 1 | 9 | 9 |
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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9660 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 6721 & 5250 \\ 8295 & 6091 \end{array}\right),\left(\begin{array}{rr} 4829 & 0 \\ 0 & 9659 \end{array}\right),\left(\begin{array}{rr} 3863 & 0 \\ 0 & 9659 \end{array}\right),\left(\begin{array}{rr} 6899 & 0 \\ 0 & 9659 \end{array}\right),\left(\begin{array}{rr} 9655 & 6 \\ 9654 & 7 \end{array}\right),\left(\begin{array}{rr} 419 & 4410 \\ 420 & 4409 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5494 & 8715 \\ 6545 & 4409 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9660])$ is a degree-$74457541509120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9660\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 450800cs
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 3220c2, its twist by $140$.
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.