Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-33080x+2447547\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-33080xz^2+2447547z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-529275x+156113750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(89, 405\right)\) | \(\left(-1, 1575\right)\) |
$\hat{h}(P)$ | ≈ | $0.62125428243289787126785382599$ | $0.62140419068760066890716397014$ |
Torsion generators
\( \left(-211, 105\right) \)
Integral points
\( \left(-211, 105\right) \), \( \left(-181, 1665\right) \), \( \left(-181, -1485\right) \), \( \left(-127, 2205\right) \), \( \left(-127, -2079\right) \), \( \left(-67, 2121\right) \), \( \left(-67, -2055\right) \), \( \left(-51, 2025\right) \), \( \left(-51, -1975\right) \), \( \left(-1, 1575\right) \), \( \left(-1, -1575\right) \), \( \left(45, 1001\right) \), \( \left(45, -1047\right) \), \( \left(69, 665\right) \), \( \left(69, -735\right) \), \( \left(89, 405\right) \), \( \left(89, -495\right) \), \( \left(113, 321\right) \), \( \left(113, -435\right) \), \( \left(125, 441\right) \), \( \left(125, -567\right) \), \( \left(149, 825\right) \), \( \left(149, -975\right) \), \( \left(269, 3465\right) \), \( \left(269, -3735\right) \), \( \left(419, 7665\right) \), \( \left(419, -8085\right) \), \( \left(489, 9905\right) \), \( \left(489, -10395\right) \), \( \left(989, 30105\right) \), \( \left(989, -31095\right) \), \( \left(999, 30575\right) \), \( \left(999, -31575\right) \), \( \left(3149, 174825\right) \), \( \left(3149, -177975\right) \), \( \left(5165, 368361\right) \), \( \left(5165, -373527\right) \), \( \left(65399, 16691775\right) \), \( \left(65399, -16757175\right) \), \( \left(38036069, 234562307265\right) \), \( \left(38036069, -234600343335\right) \)
Invariants
Conductor: | \( 116550 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 37$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-253761984000000 $ | = | $-1 \cdot 2^{12} \cdot 3^{7} \cdot 5^{6} \cdot 7^{2} \cdot 37 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -\frac{351447414193}{22278144} \) | = | $-1 \cdot 2^{-12} \cdot 3^{-1} \cdot 7^{-2} \cdot 37^{-1} \cdot 7057^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
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: | $1.5175305025447486839344663068\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.16350540199364365093646402173\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9888559317939599\dots$ | |||
Szpiro ratio: | $3.6805496219801976\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.32081165398667212125087449142\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.54516727769035551302560454888\dots$ | 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: | $ 384 $ = $ ( 2^{2} \cdot 3 )\cdot2^{2}\cdot2^{2}\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: | $2$ | 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^{(2)}(E,1)/2! $ ≈ $ 16.790017541304415546244625164 $ | 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
$\displaystyle 16.790017541 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.545167 \cdot 0.320812 \cdot 384}{2^2} \approx 16.790017541$
Modular invariants
Modular form 116550.2.a.df
For more coefficients, see the Downloads section to the right.
Modular degree: | 491520 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$3$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$5$ | $4$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
$7$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$37$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3108 = 2^{2} \cdot 3 \cdot 7 \cdot 37 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1262 & 1 \\ 923 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1333 & 4 \\ 2666 & 9 \end{array}\right),\left(\begin{array}{rr} 3105 & 4 \\ 3104 & 5 \end{array}\right),\left(\begin{array}{rr} 781 & 2332 \\ 776 & 2331 \end{array}\right),\left(\begin{array}{rr} 2074 & 1 \\ 2071 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[3108])$ is a degree-$1410626617344$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3108\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 116550eh
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1554m1, its twist by $-15$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-111}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.2.2175600.3 | \(\Z/4\Z\) | Not in database |
$8$ | 8.0.58318192870560000.28 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | add | nonsplit | ord | ord | ord | ord | ord | ord | ord | split | ord | ord | ord |
$\lambda$-invariant(s) | 5 | - | - | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 2 | 2 | 2 |
$\mu$-invariant(s) | 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.