Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-48033x-3047391\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-48033xz^2-3047391z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-3890700x-2233220112\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-120, 987)$ | $4.7198654370409774446100268120$ | $\infty$ |
| $(-71, 0)$ | $0$ | $2$ |
Integral points
\((-120,\pm 987)\), \( \left(-71, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 3264 \) | = | $2^{6} \cdot 3 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
| Discriminant: | $\Delta$ | = | $3038569102835712$ | = | $2^{36} \cdot 3^{2} \cdot 17^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
| j-invariant: | $j$ | = | \( \frac{46753267515625}{11591221248} \) | = | $2^{-18} \cdot 3^{-2} \cdot 5^{6} \cdot 11^{3} \cdot 17^{-3} \cdot 131^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6811470915213376186740260401$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.64142632068141965454817785791$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0866579960559595$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.432472735708098$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
| Mordell-Weil rank: | $r$ | = | $ 1$ | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.7198654370409774446100268120$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
| Real period: | $\Omega$ | ≈ | $0.32834090721370750857074646876$ | 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: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{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: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
| Special value: | $ L'(E,1)$ | ≈ | $3.0994497990493132273380063591 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
BSD formula
$\displaystyle 3.099449799 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.328341 \cdot 4.719865 \cdot 8}{2^2} \approx 3.099449799$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 13824 | 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 at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{26}^{*}$ | additive | 1 | 6 | 36 | 18 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $17$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 | 8.6.0.4 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 408 = 2^{3} \cdot 3 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 370 & 3 \\ 21 & 400 \end{array}\right),\left(\begin{array}{rr} 203 & 396 \\ 402 & 335 \end{array}\right),\left(\begin{array}{rr} 397 & 12 \\ 396 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 358 & 399 \end{array}\right),\left(\begin{array}{rr} 137 & 12 \\ 340 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 327 & 394 \\ 38 & 101 \end{array}\right)$.
The torsion field $K:=\Q(E[408])$ is a degree-$60162048$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/408\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 17 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 64 = 2^{6} \) |
| $17$ | nonsplit multiplicative | $18$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 3264a
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 102c3, its twist by $8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.4352.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.30233088.6 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.5473632256.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.8008266092544.7 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1534132224.11 | \(\Z/12\Z\) | not in database |
| $12$ | 12.0.8226356490141696.32 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.48300968137817343423391699486245912576.1 | \(\Z/18\Z\) | 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 | add | nonsplit | ss | ord | ss | ord | nonsplit | ord | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 3,1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 1 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.