Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-2722542176x-215546840077824\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-2722542176xz^2-215546840077824z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-220525916283x-157134307994482518\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{9487608278268010852907930835092951938790490401842762}{120640662827229859611443151772262358409414076169}, \frac{315611754600436207087078339500559250490010238847754034264678186499504703208038}{41902561481186394837667337164953370096250091442597601180943047352577947}\right)\) |
$\hat{h}(P)$ | ≈ | $119.43082517223356056869143952$ |
Torsion generators
\( \left(74849, 0\right) \)
Integral points
\( \left(74849, 0\right) \)
Invariants
Conductor: | \( 443760 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 43^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-18779551213226359220122060800000 $ | = | $-1 \cdot 2^{14} \cdot 3^{22} \cdot 5^{5} \cdot 43^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{86193969101536367161}{725294740213012500} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-22} \cdot 5^{-5} \cdot 43^{-2} \cdot 4417321^{3}$ | 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: | $4.6835516441093113460727243406\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.1098044057025848249190709625\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0397516832930858\dots$ | |||
Szpiro ratio: | $6.116381849365313\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $119.43082517223356056869143952\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.0091818462775872450933105840832\dots$ | 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: | $ 16 $ = $ 2\cdot2\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) $ ≈ $ 4.3863819101473830718180873935 $ | 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
$\displaystyle 4.386381910 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.009182 \cdot 119.430825 \cdot 16}{2^2} \approx 4.386381910$
Modular invariants
Modular form 443760.2.a.f
For more coefficients, see the Downloads section to the right.
Modular degree: | 936714240 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{6}^{*}$ | Additive | -1 | 4 | 14 | 2 |
$3$ | $2$ | $I_{22}$ | Non-split multiplicative | 1 | 1 | 22 | 22 |
$5$ | $1$ | $I_{5}$ | Non-split multiplicative | 1 | 1 | 5 | 5 |
$43$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2580 = 2^{2} \cdot 3 \cdot 5 \cdot 43 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1936 & 649 \\ 645 & 1936 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1721 & 4 \\ 862 & 9 \end{array}\right),\left(\begin{array}{rr} 1034 & 1 \\ 2063 & 0 \end{array}\right),\left(\begin{array}{rr} 1981 & 4 \\ 1382 & 9 \end{array}\right),\left(\begin{array}{rr} 2577 & 4 \\ 2576 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[2580])$ is a degree-$615165788160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2580\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 443760.f
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1290.b2, its twist by $172$.
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.