Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-948119x-401927292\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-948119xz^2-401927292z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15169907x-25738516594\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(1292, 22344\right)\)
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| $\hat{h}(P)$ | ≈ | $1.6638179689958272347419915734$ |
Torsion generators
\( \left(\frac{4563}{4}, -\frac{4563}{8}\right) \)
Integral points
\( \left(1292, 22344\right) \), \( \left(1292, -23636\right) \), \( \left(20672, 2958414\right) \), \( \left(20672, -2979086\right) \)
Invariants
| Conductor: | \( 1045 \) | = | $5 \cdot 11 \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-15323275604248046875 $ | = | $-1 \cdot 5^{16} \cdot 11^{4} \cdot 19^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{94256762600623910012361}{15323275604248046875} \) | = | $-1 \cdot 3^{3} \cdot 5^{-16} \cdot 11^{-4} \cdot 19^{-3} \cdot 61^{3} \cdot 431^{3} \cdot 577^{3}$ | 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: | $2.4089712693847815425058418295\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $2.4089712693847815425058418295\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $1.6638179689958272347419915734\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.075822212874574098349463719319\dots$ | 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: | $ 96 $ = $ 2^{4}\cdot2\cdot3 $ | 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: | $1$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L'(E,1) $ ≈ $ 3.0277046455138353542434232564 $ | 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
$\displaystyle 3.027704646 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.075822 \cdot 1.663818 \cdot 96}{2^2} \approx 3.027704646$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 17664 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $5$ | $16$ | $I_{16}$ | Split multiplicative | -1 | 1 | 16 | 16 |
| $11$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
| $19$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
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 | 4.12.0.8 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1672 = 2^{3} \cdot 11 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 761 & 8 \\ 1372 & 33 \end{array}\right),\left(\begin{array}{rr} 624 & 3 \\ 181 & 2 \end{array}\right),\left(\begin{array}{rr} 1665 & 8 \\ 1664 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 635 & 628 \\ 674 & 1469 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1666 & 1667 \end{array}\right),\left(\begin{array}{rr} 1467 & 1466 \\ 1058 & 219 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[1672])$ is a degree-$4815382$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1672\Z)$.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ord | ss | split | ss | nonsplit | ord | ord | split | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 1 | 3,3 | 2 | 3,1 | 1 | 3 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 1 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 1045b
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-19}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{19}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | Not in database |
| $4$ | 4.0.109744.1 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $4$ | \(\Q(i, \sqrt{19})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.0.192699928576.2 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.4.45141114468499456.5 | \(\Z/8\Z\) | Not in database |
| $8$ | 8.0.346384039936.3 | \(\Z/8\Z\) | Not in database |
| $8$ | 8.2.20012416875.1 | \(\Z/6\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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.