Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-4500x+117296\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-4500xz^2+117296z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-71995x+7434966\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(37, 2\right)\) |
$\hat{h}(P)$ | ≈ | $1.4952964202786590582634983739$ |
Torsion generators
\( \left(\frac{155}{4}, -\frac{159}{8}\right) \)
Integral points
\( \left(37, 2\right) \), \( \left(37, -40\right) \), \( \left(69, 328\right) \), \( \left(69, -398\right) \)
Invariants
Conductor: | \( 5929 \) | = | $7^{2} \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $607645423 $ | = | $7^{3} \cdot 11^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( 16581375 \) | = | $3^{3} \cdot 5^{3} \cdot 17^{3}$ | 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: | \(\Z[\sqrt{-7}]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
Faltings height: | $0.74638283762353550961527613999\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.93904233603947808869203383485\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.1980441775334598\dots$ | |||
Szpiro ratio: | $4.241545044127054\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.4952964202786590582634983739\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.5422492151378039418624898585\dots$ | 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 $ = $ 2\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'(E,1) $ ≈ $ 2.3061197305731297548267891647 $ | 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
$\displaystyle 2.306119731 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.542249 \cdot 1.495296 \cdot 4}{2^2} \approx 2.306119731$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 2560 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$7$ | $2$ | $III$ | Additive | -1 | 2 | 3 | 0 |
$11$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 7 and 14.
Its isogeny class 5929e
consists of 4 curves linked by isogenies of
degrees dividing 14.
Twists
The minimal quadratic twist of this elliptic curve is 49a2, its twist by $-11$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.0.166012.1 | \(\Z/4\Z\) | Not in database |
$6$ | 6.6.22370117.1 | \(\Z/14\Z\) | Not in database |
$8$ | 8.4.112885695053824.10 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.440959746304.7 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.7055355940864.34 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.440959746304.8 | \(\Z/8\Z\) | Not in database |
$8$ | 8.2.3767105332683.1 | \(\Z/6\Z\) | Not in database |
$12$ | 12.12.2049729063295750144.1 | \(\Z/2\Z \oplus \Z/14\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$20$ | 20.0.163898699393368601103605933.1 | \(\Z/22\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | ord | ss | ss | add | add | ss | ss | ss | ord | ord | ss | ord | ss | ord | ss |
$\lambda$-invariant(s) | ? | 3,1 | 1,1 | - | - | 1,1 | 1,3 | 1,1 | 1 | 1 | 1,1 | 1 | 1,1 | 1 | 1,1 |
$\mu$-invariant(s) | ? | 0,0 | 0,0 | - | - | 0,0 | 0,0 | 0,0 | 0 | 0 | 0,0 | 0 | 0,0 | 0 | 0,0 |
An entry ? indicates that the invariants have not yet been computed.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.