Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-5378257x+4817149415\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-5378257xz^2+4817149415z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-6970220451x+224769833779230\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(\frac{2955}{4}, \frac{279575}{8}\right)\) | \(\left(\frac{94369}{64}, \frac{4566613}{512}\right)\) |
$\hat{h}(P)$ | ≈ | $1.2086505595564586116219331499$ | $2.0008610333942245327380308734$ |
Torsion generators
\( \left(-2679, 1339\right) \)
Integral points
\( \left(-2679, 1339\right) \), \( \left(1165, 10949\right) \), \( \left(1165, -12115\right) \), \( \left(1353, 3523\right) \), \( \left(1353, -4877\right) \), \( \left(6925, 543965\right) \), \( \left(6925, -550891\right) \)
Invariants
Conductor: | \( 20181 \) | = | $3 \cdot 7 \cdot 31^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-69944692694810747751 $ | = | $-1 \cdot 3^{2} \cdot 7^{10} \cdot 31^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -\frac{19385548183592137}{78810594471} \) | = | $-1 \cdot 3^{-2} \cdot 7^{-10} \cdot 31^{-1} \cdot 353^{3} \cdot 761^{3}$ | 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: | $2.6640804252432204071179962684\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.94708682300064728415341410613\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $0.9761213698736007\dots$ | |||
Szpiro ratio: | $5.862723476516964\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $2.3803385531671165151387188835\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.19585543275585420904599871884\dots$ | 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: | $ 80 $ = $ 2\cdot( 2 \cdot 5 )\cdot2^{2} $ | 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! $ ≈ $ 9.3240447487197897526531229814 $ | 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
$\displaystyle 9.324044749 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.195855 \cdot 2.380339 \cdot 80}{2^2} \approx 9.324044749$
Modular invariants
Modular form 20181.2.a.o
For more coefficients, see the Downloads section to the right.
Modular degree: | 921600 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$7$ | $10$ | $I_{10}$ | Split multiplicative | -1 | 1 | 10 | 10 |
$31$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 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 \( 868 = 2^{2} \cdot 7 \cdot 31 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 745 & 4 \\ 622 & 9 \end{array}\right),\left(\begin{array}{rr} 254 & 1 \\ 307 & 0 \end{array}\right),\left(\begin{array}{rr} 865 & 4 \\ 864 & 5 \end{array}\right),\left(\begin{array}{rr} 221 & 652 \\ 216 & 651 \end{array}\right)$.
The torsion field $K:=\Q(E[868])$ is a degree-$14399078400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/868\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 20181.o
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 651.c2, its twist by $-31$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-31}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.2.24304.1 | \(\Z/4\Z\) | Not in database |
$8$ | 8.0.567647723776.4 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.56360033758224.3 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | ord | split | ord | split | ord | ord | ord | ord | ord | ord | add | ord | ord | ord | ord |
$\lambda$-invariant(s) | 17 | 5 | 8 | 3 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.