Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-858195x-123529379\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-858195xz^2-123529379z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-13731123x-7919611378\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-243, 8525\right)\) | \(\left(-187, 5606\right)\) |
$\hat{h}(P)$ | ≈ | $1.9648793582261344764535881878$ | $2.9582540944240401999866982764$ |
Torsion generators
\( \left(-\frac{3373}{4}, \frac{3373}{8}\right) \)
Integral points
\( \left(-823, 5393\right) \), \( \left(-823, -4570\right) \), \( \left(-243, 8525\right) \), \( \left(-243, -8282\right) \), \( \left(-187, 5606\right) \), \( \left(-187, -5419\right) \), \( \left(1283, 29126\right) \), \( \left(1283, -30409\right) \), \( \left(2127, 86518\right) \), \( \left(2127, -88645\right) \), \( \left(62183, 15473366\right) \), \( \left(62183, -15535549\right) \)
Invariants
Conductor: | \( 101430 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $33836806552378548600 $ | = | $2^{3} \cdot 3^{12} \cdot 5^{2} \cdot 7^{12} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{815016062816929}{394524156600} \) | = | $2^{-3} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{-6} \cdot 23^{-1} \cdot 29^{3} \cdot 3221^{3}$ | 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: | $2.4409150638518660621561155534\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.91865384499015456390581656322\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $0.9487456883116873\dots$ | |||
Szpiro ratio: | $4.5632685761828835\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $5.7083467625166374639956788792\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.16465024483733574079085399393\dots$ | 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: | $ 32 $ = $ 1\cdot2^{2}\cdot2\cdot2^{2}\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);
|
Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 7.5190455365182174207149717719 $ | 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
$\displaystyle 7.519045537 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.164650 \cdot 5.708347 \cdot 32}{2^2} \approx 7.519045537$
Modular invariants
Modular form 101430.2.a.a
For more coefficients, see the Downloads section to the right.
Modular degree: | 2654208 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$3$ | $4$ | $I_{6}^{*}$ | Additive | -1 | 2 | 12 | 6 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$7$ | $4$ | $I_{6}^{*}$ | Additive | -1 | 2 | 12 | 6 |
$23$ | $1$ | $I_{1}$ | 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 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 19320 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 19270 & 19311 \end{array}\right),\left(\begin{array}{rr} 9250 & 3 \\ 11733 & 19312 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 15457 & 12 \\ 15462 & 73 \end{array}\right),\left(\begin{array}{rr} 19309 & 12 \\ 19308 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12871 & 19318 \\ 3216 & 19319 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 9633 & 19312 \end{array}\right),\left(\begin{array}{rr} 4026 & 817 \\ 13685 & 10466 \end{array}\right),\left(\begin{array}{rr} 16559 & 19308 \\ 2754 & 19247 \end{array}\right)$.
The torsion field $K:=\Q(E[19320])$ is a degree-$198553444024320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19320\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 101430.a
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 4830.bf3, its twist by $21$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{46}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{21}) \) | \(\Z/6\Z\) | Not in database |
$4$ | 4.0.8114400.1 | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{21}, \sqrt{46})\) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.0.539918229375.1 | \(\Z/6\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 |
$18$ | 18.6.3306851379447304569328878403463240772974400000000.1 | \(\Z/18\Z\) | 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 | nonsplit | add | nonsplit | add | ord | ord | ss | ord | split | ss | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 5 | - | 4 | - | 2 | 2 | 2,2 | 4 | 3 | 2,2 | 2 | 2 | 2 | 2 | 2 |
$\mu$-invariant(s) | 1 | - | 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.