Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-27608463x-55828700901\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-27608463xz^2-55828700901z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-441735411x-3573478593074\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(-3035, 1428\right)\)
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| $\hat{h}(P)$ | ≈ | $5.2216925311795491589695449259$ |
Torsion generators
\( \left(-\frac{12141}{4}, \frac{12141}{8}\right) \)
Integral points
\( \left(-3035, 1607\right) \), \( \left(-3035, 1428\right) \), \( \left(134235, 49075632\right) \), \( \left(134235, -49209867\right) \)
Invariants
| Conductor: | \( 404586 \) | = | $2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $1560042231714109962 $ | = | $2 \cdot 3^{11} \cdot 7 \cdot 13^{6} \cdot 19^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{661397832743623417}{443352042} \) | = | $2^{-1} \cdot 3^{-5} \cdot 7^{-1} \cdot 13^{3} \cdot 19^{-4} \cdot 67021^{3}$ | 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: | $2.8070825854320853246135258033\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $0.97530176236726211088915946405\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.0038346664696574\dots$ | |||
| Szpiro ratio: | $4.880827082100609\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $5.2216925311795491589695449259\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.065862621692976190039977904984\dots$ | 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: | $ 32 $ = $ 1\cdot2\cdot1\cdot2^{2}\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) $ ≈ $ 2.7513148782249434002007391837 $ | 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
$\displaystyle 2.751314878 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.065863 \cdot 5.221693 \cdot 32}{2^2} \approx 2.751314878$
Modular invariants
Modular form 404586.2.a.i
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23592960 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_{5}^{*}$ | Additive | -1 | 2 | 11 | 5 |
| $7$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $4$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
| $19$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
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 \( 41496 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 41490 & 41491 \end{array}\right),\left(\begin{array}{rr} 30848 & 19149 \\ 18083 & 6382 \end{array}\right),\left(\begin{array}{rr} 23544 & 5993 \\ 14755 & 21126 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 35111 & 0 \\ 0 & 41495 \end{array}\right),\left(\begin{array}{rr} 39313 & 31928 \\ 7228 & 3225 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 41489 & 8 \\ 41488 & 9 \end{array}\right),\left(\begin{array}{rr} 2003 & 40300 \\ 37154 & 35517 \end{array}\right),\left(\begin{array}{rr} 16420 & 35113 \\ 36959 & 9582 \end{array}\right)$.
The torsion field $K:=\Q(E[41496])$ is a degree-$9991811456040960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/41496\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 404586.i
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 798.c1, its twist by $-39$.
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.