Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-4730848x-3936856594\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-4730848xz^2-3936856594z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-6131178387x-183659587702866\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(3145, 109307\right)\) |
$\hat{h}(P)$ | ≈ | $5.0224796055803294268187027594$ |
Torsion generators
\( \left(-1335, 667\right) \), \( \left(-1175, 587\right) \)
Integral points
\( \left(-1335, 667\right) \), \( \left(-1175, 587\right) \), \( \left(3145, 109307\right) \), \( \left(3145, -112453\right) \)
Invariants
Conductor: | \( 46410 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $82207646338733697600 $ | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{6} \cdot 17^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{11709559667189768059461241}{82207646338733697600} \) | = | $2^{-6} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 13^{-6} \cdot 17^{-6} \cdot 79^{3} \cdot 109^{3} \cdot 26371^{3}$ | 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: | $2.6541967314706582988809298555\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $2.6541967314706582988809298555\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $5.0224796055803294268187027594\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.10241146407749059481791578232\dots$ | 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: | $ 192 $ = $ 2\cdot2\cdot2\cdot2\cdot( 2 \cdot 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: | $4$ | 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'(E,1) $ ≈ $ 6.1723138764818284596006598792 $ | 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
$\displaystyle 6.172313876 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.102411 \cdot 5.022480 \cdot 192}{4^2} \approx 6.172313876$
Modular invariants
Modular form 46410.2.a.be
For more coefficients, see the Downloads section to the right.
Modular degree: | 1824768 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$3$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$5$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$7$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$13$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$17$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
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 |
---|---|---|
$2$ | 2Cs | 2.6.0.1 |
$3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 185640 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 139231 & 12 \\ 92826 & 73 \end{array}\right),\left(\begin{array}{rr} 185629 & 12 \\ 185628 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 92827 & 6 \\ 185634 & 185635 \end{array}\right),\left(\begin{array}{rr} 106081 & 12 \\ 79566 & 73 \end{array}\right),\left(\begin{array}{rr} 157081 & 12 \\ 14286 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 148519 & 6 \\ 111378 & 185635 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 185624 & 185633 \end{array}\right),\left(\begin{array}{rr} 65521 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 123761 & 6 \\ 30940 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[185640])$ is a degree-$381441784707809280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/185640\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 46410be
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$3$ | 3.1.297675.3 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$4$ | \(\Q(\sqrt{10}, \sqrt{-51})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-10}, \sqrt{-91})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{51}, \sqrt{91})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$6$ | 6.0.265831216875.4 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$8$ | 8.0.17318914560000.12 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$18$ | 18.0.77334001643176291132352643770588904079275233936683200000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | nonsplit | split | split | split | ss | split | nonsplit | ord | ss | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 7 | 2 | 2 | 2 | 1,1 | 2 | 1 | 1 | 1,1 | 3 | 1 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | 0 | 1 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.