Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-240152x+45242651\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-240152xz^2+45242651z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-3842427x+2891687254\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(81, 5089\right)\) | \(\left(261, 409\right)\) |
$\hat{h}(P)$ | ≈ | $0.30807960444276467675738308583$ | $0.70292700705274157485213662840$ |
Torsion generators
\( \left(\frac{1179}{4}, -\frac{1183}{8}\right) \)
Integral points
\( \left(-549, 3649\right) \), \( \left(-549, -3101\right) \), \( \left(-339, 9529\right) \), \( \left(-339, -9191\right) \), \( \left(-299, 9649\right) \), \( \left(-299, -9351\right) \), \( \left(-159, 8989\right) \), \( \left(-159, -8831\right) \), \( \left(-9, 6889\right) \), \( \left(-9, -6881\right) \), \( \left(81, 5089\right) \), \( \left(81, -5171\right) \), \( \left(201, 2149\right) \), \( \left(201, -2351\right) \), \( \left(261, 409\right) \), \( \left(261, -671\right) \), \( \left(271, -41\right) \), \( \left(271, -231\right) \), \( \left(301, 249\right) \), \( \left(301, -551\right) \), \( \left(309, 529\right) \), \( \left(309, -839\right) \), \( \left(351, 1849\right) \), \( \left(351, -2201\right) \), \( \left(405, 3577\right) \), \( \left(405, -3983\right) \), \( \left(505, 6997\right) \), \( \left(505, -7503\right) \), \( \left(601, 10549\right) \), \( \left(601, -11151\right) \), \( \left(651, 12499\right) \), \( \left(651, -13151\right) \), \( \left(1449, 51601\right) \), \( \left(1449, -53051\right) \), \( \left(1701, 66649\right) \), \( \left(1701, -68351\right) \), \( \left(3501, 203449\right) \), \( \left(3501, -206951\right) \), \( \left(10801, 1115949\right) \), \( \left(10801, -1126751\right) \), \( \left(17751, 2355199\right) \), \( \left(17751, -2372951\right) \), \( \left(222951, 105160399\right) \), \( \left(222951, -105383351\right) \)
Invariants
Conductor: | \( 32490 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $4500189900000000 $ | = | $2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{306331959547531}{900000000} \) | = | $2^{-8} \cdot 3^{-2} \cdot 5^{-8} \cdot 67411^{3}$ | 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: | $1.8737411983511113325099363279\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.58832530922544637181005685147\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.021073372079632\dots$ | |||
Szpiro ratio: | $4.6955575986242675\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.20231695556169541507808413978\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.43721416254864402281971971727\dots$ | 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: | $ 512 $ = $ 2^{3}\cdot2^{2}\cdot2^{3}\cdot2 $ | 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);
|
Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 11.322347301798129739086561410 $ | 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
$\displaystyle 11.322347302 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.437214 \cdot 0.202317 \cdot 512}{2^2} \approx 11.322347302$
Modular invariants
Modular form 32490.2.a.bo
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) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$3$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
$5$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$19$ | $2$ | $III$ | Additive | 1 | 2 | 3 | 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 |
---|---|---|
$2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 228 = 2^{2} \cdot 3 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 172 & 61 \\ 57 & 172 \end{array}\right),\left(\begin{array}{rr} 160 & 1 \\ 131 & 0 \end{array}\right),\left(\begin{array}{rr} 225 & 4 \\ 224 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 77 & 4 \\ 154 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[228])$ is a degree-$47278080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/228\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 32490.bo
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 10830.i1, its twist by $-3$.
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 |
$4$ | 4.4.246924.1 | \(\Z/4\Z\) | Not in database |
$8$ | 8.0.249739107434496.55 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.8.975543388416.1 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | split | ord | ord | ord | ord | add | ord | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | 3 | - | 3 | 2 | 2 | 2 | 2 | - | 2 | 2 | 2,2 | 2 | 2 | 2 | 2 |
$\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.