Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-91764612x-338322994479\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-91764612xz^2-338322994479z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1468233795x-21654139880450\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{135303781163326427235295231231993942878175053304}{11735867812399723604593347930068995846236721}, \frac{14598027428843134255948673538105596225207017602759477494848062287008293}{40204328583013950560119122671929264483412365461521076806104390231}\right)\) |
$\hat{h}(P)$ | ≈ | $108.11096678563828121087475576$ |
Integral points
None
Invariants
Conductor: | \( 11025 \) | = | $3^{2} \cdot 5^{2} \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-525317491125 $ | = | $-1 \cdot 3^{6} \cdot 5^{3} \cdot 7^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -162677523113838677 \) | = | $-1 \cdot 7 \cdot 137^{3} \cdot 2083^{3}$ | 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: | $2.7927459473690799612509332590\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.54380689222295781849955231160\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0734398000116747\dots$ | |||
Szpiro ratio: | $7.15711027811198\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $108.11096678563828121087475576\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.024389379456961113991055055259\dots$ | 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);
|
Tamagawa product: | $ 2 $ = $ 1\cdot2\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)
|
Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
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) $ ≈ $ 5.2735187847877032287516497983 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 5.273518785 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.024389 \cdot 108.110967 \cdot 2}{1^2} \approx 5.273518785$
Modular invariants
Modular form 11025.2.a.bg
For more coefficients, see the Downloads section to the right.
Modular degree: | 372960 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $1$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$5$ | $2$ | $III$ | Additive | -1 | 2 | 3 | 0 |
$7$ | $1$ | $IV^{*}$ | Additive | 1 | 2 | 8 | 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 |
---|---|---|
$37$ | 37B.8.2 | 37.114.4.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15540 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 37 \), index $2736$, genus $97$, and generators
$\left(\begin{array}{rr} 1 & 5208 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 5328 & 1 \end{array}\right),\left(\begin{array}{rr} 4441 & 0 \\ 0 & 8881 \end{array}\right),\left(\begin{array}{rr} 15208 & 8103 \\ 9657 & 6439 \end{array}\right),\left(\begin{array}{rr} 5181 & 10360 \\ 5180 & 5181 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10360 & 1 \end{array}\right),\left(\begin{array}{rr} 5179 & 0 \\ 0 & 15539 \end{array}\right),\left(\begin{array}{rr} 1 & 8436 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10465 & 8214 \\ 5217 & 11143 \end{array}\right),\left(\begin{array}{rr} 10435 & 10434 \\ 6771 & 12655 \end{array}\right),\left(\begin{array}{rr} 1 & 7770 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7771 & 7770 \\ 7770 & 7771 \end{array}\right),\left(\begin{array}{rr} 6217 & 0 \\ 0 & 12433 \end{array}\right),\left(\begin{array}{rr} 15455 & 12432 \\ 10212 & 7151 \end{array}\right),\left(\begin{array}{rr} 5293 & 9324 \\ 5328 & 1 \end{array}\right),\left(\begin{array}{rr} 7105 & 8436 \\ 7104 & 7105 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 7770 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[15540])$ is a degree-$2969740247040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15540\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
37.
Its isogeny class 11025bg
consists of 2 curves linked by isogenies of
degree 37.
Twists
The minimal quadratic twist of this elliptic curve is 1225h2, its twist by $21$.
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 |
---|---|---|---|
$3$ | 3.1.980.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.19208000.2 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | 8.2.82046671875.4 | \(\Z/3\Z\) | 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 | ord | add | add | add | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | ? | - | - | - | 3,3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | ? | - | - | - | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 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.