Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-590x-2788\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-590xz^2-2788z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-9435x-187850\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-6, 25\right)\) | \(\left(39, 160\right)\) | \(\left(183, 2356\right)\) |
$\hat{h}(P)$ | ≈ | $0.95251018469599268210849459822$ | $1.7590133221900932839090367475$ | $2.4856526401905807575106831548$ |
Torsion generators
\( \left(-21, 10\right) \)
Integral points
\( \left(-21, 10\right) \), \( \left(-20, 36\right) \), \( \left(-20, -17\right) \), \( \left(-12, 55\right) \), \( \left(-12, -44\right) \), \( \left(-6, 25\right) \), \( \left(-6, -20\right) \), \( \left(-5, 6\right) \), \( \left(-5, -2\right) \), \( \left(28, 31\right) \), \( \left(28, -60\right) \), \( \left(30, 61\right) \), \( \left(30, -92\right) \), \( \left(39, 160\right) \), \( \left(39, -200\right) \), \( \left(60, 388\right) \), \( \left(60, -449\right) \), \( \left(64, 435\right) \), \( \left(64, -500\right) \), \( \left(123, 1270\right) \), \( \left(123, -1394\right) \), \( \left(183, 2356\right) \), \( \left(183, -2540\right) \), \( \left(354, 6460\right) \), \( \left(354, -6815\right) \), \( \left(714, 18700\right) \), \( \left(714, -19415\right) \), \( \left(744, 19900\right) \), \( \left(744, -20645\right) \), \( \left(48939, 10801810\right) \), \( \left(48939, -10850750\right) \)
Invariants
Conductor: | \( 455175 \) | = | $3^{2} \cdot 5^{2} \cdot 7 \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $9401639625 $ | = | $3^{7} \cdot 5^{3} \cdot 7 \cdot 17^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{50653}{21} \) | = | $3^{-1} \cdot 7^{-1} \cdot 37^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $0.61093981067199576011560165406\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.0490291477846381992945944522\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.7545703377951682\dots$ | |||
Szpiro ratio: | $2.3604044502342956\dots$ |
BSD invariants
Analytic rank: | $3$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.6538237924103767954060386860\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.0049673156667045364540243493\dots$ | 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^{2}\cdot2\cdot1\cdot2 $ | 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^{(3)}(E,1)/3! $ ≈ $ 14.687893954311178576084626343 $ | 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
$\displaystyle 14.687893954 \approx L^{(3)}(E,1)/3! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.004967 \cdot 3.653824 \cdot 16}{2^2} \approx 14.687893954$
Modular invariants
Modular form 455175.2.a.bb
For more coefficients, see the Downloads section to the right.
Modular degree: | 327680 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$5$ | $2$ | $III$ | Additive | -1 | 2 | 3 | 0 |
$7$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$17$ | $2$ | $III$ | Additive | 1 | 2 | 3 | 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7140 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 4288 & 1 \\ 4283 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 5357 & 1786 \\ 1784 & 5355 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5102 & 1 \\ 1019 & 0 \end{array}\right),\left(\begin{array}{rr} 4762 & 1 \\ 4759 & 0 \end{array}\right),\left(\begin{array}{rr} 5044 & 1 \\ 5879 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 7137 & 4 \\ 7136 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[7140])$ is a degree-$29108805304320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7140\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 455175.bb
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 151725.cu2, its twist by $-3$.
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.