Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-35736950x+85662796500\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-35736950xz^2+85662796500z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-46315087875x+3997378159818750\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 139650 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-250161742630059750000000 $ | = | $-1 \cdot 2^{7} \cdot 3^{11} \cdot 5^{9} \cdot 7^{7} \cdot 19^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{21966350325866981}{1088685940608} \) | = | $-1 \cdot 2^{-7} \cdot 3^{-11} \cdot 7^{-1} \cdot 19^{-3} \cdot 280061^{3}$ | 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: | $3.2509815419938444951937799135\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.0709480331406125616905340419\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9697990043345774\dots$ | |||
Szpiro ratio: | $5.391359033730491\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.097482323526021725971838350807\dots$ | 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: | $ 12 $ = $ 1\cdot1\cdot2\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: | $1$ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 1.1697878823122607116620602097 $ | 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
$\displaystyle 1.169787882 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.097482 \cdot 1.000000 \cdot 12}{1^2} \approx 1.169787882$
Modular invariants
Modular form 139650.2.a.g
For more coefficients, see the Downloads section to the right.
Modular degree: | 31933440 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{7}$ | Non-split multiplicative | 1 | 1 | 7 | 7 |
$3$ | $1$ | $I_{11}$ | Non-split multiplicative | 1 | 1 | 11 | 11 |
$5$ | $2$ | $III^{*}$ | Additive | -1 | 2 | 9 | 0 |
$7$ | $2$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$19$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
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 \( 15960 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 9577 & 2 \\ 9577 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 15959 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3991 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15959 & 2 \\ 15958 & 3 \end{array}\right),\left(\begin{array}{rr} 13681 & 2 \\ 13681 & 3 \end{array}\right),\left(\begin{array}{rr} 5321 & 2 \\ 5321 & 3 \end{array}\right),\left(\begin{array}{rr} 4201 & 2 \\ 4201 & 3 \end{array}\right),\left(\begin{array}{rr} 7981 & 2 \\ 7981 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[15960])$ is a degree-$4392005035622400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15960\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 139650.g consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 19950.cd1, its twist by $-35$.
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 |
---|---|---|---|
$3$ | 3.1.15960.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.4065356736000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | 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 | nonsplit | nonsplit | add | add | ord | ord | ord | split | ord | ord | ord | ord | ss | ord | ord |
$\lambda$-invariant(s) | 4 | 4 | - | - | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 |
$\mu$-invariant(s) | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.