Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-7980x+157647\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-7980xz^2+157647z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-127675x+9961750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(329, 5585\right)\) |
$\hat{h}(P)$ | ≈ | $0.13845885765230944890086936276$ |
Integral points
\( \left(79, 85\right) \), \( \left(79, -165\right) \), \( \left(99, 525\right) \), \( \left(99, -625\right) \), \( \left(145, 1353\right) \), \( \left(145, -1499\right) \), \( \left(329, 5585\right) \), \( \left(329, -5915\right) \), \( \left(1649, 66025\right) \), \( \left(1649, -67675\right) \)
Invariants
Conductor: | \( 33350 \) | = | $2 \cdot 5^{2} \cdot 23 \cdot 29$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $22052687500000 $ | = | $2^{5} \cdot 5^{9} \cdot 23^{3} \cdot 29 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{3596344921161}{1411372000} \) | = | $2^{-5} \cdot 3^{3} \cdot 5^{-3} \cdot 23^{-3} \cdot 29^{-1} \cdot 5107^{3}$ | 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: | $1.2589643846463568367340321341\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.45424542842930664943365246749\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.8822098691072361\dots$ | |||
Szpiro ratio: | $3.703145359443172\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.13845885765230944890086936276\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.61738459866463062374603041477\dots$ | 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: | $ 60 $ = $ 5\cdot2^{2}\cdot3\cdot1 $ | 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: | $1$ | 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) $ ≈ $ 5.1289419757940573883529223277 $ | 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
$\displaystyle 5.128941976 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.617385 \cdot 0.138459 \cdot 60}{1^2} \approx 5.128941976$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 190080 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$5$ | $4$ | $I_{3}^{*}$ | Additive | 1 | 2 | 9 | 3 |
$23$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$29$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 26680 = 2^{3} \cdot 5 \cdot 23 \cdot 29 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 16561 & 2 \\ 16561 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 26679 & 0 \end{array}\right),\left(\begin{array}{rr} 6671 & 2 \\ 6671 & 3 \end{array}\right),\left(\begin{array}{rr} 5801 & 2 \\ 5801 & 3 \end{array}\right),\left(\begin{array}{rr} 5337 & 2 \\ 5337 & 3 \end{array}\right),\left(\begin{array}{rr} 13341 & 2 \\ 13341 & 3 \end{array}\right),\left(\begin{array}{rr} 26679 & 2 \\ 26678 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[26680])$ is a degree-$67177248561561600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/26680\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 33350m consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 6670b1, its twist by $5$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.3.26680.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.6.18991421632000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | 8.2.15468235470000.1 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | 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 | split | ss | add | ord | ord | ord | ord | ord | split | nonsplit | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | 3 | 1,1 | - | 5 | 3 | 1 | 1 | 1 | 2 | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.