Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-2088590x-732603900\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-2088590xz^2-732603900z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-2706812667x-34172247120426\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{6}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-420, 8610\right)\) |
$\hat{h}(P)$ | ≈ | $1.3837175901661311603814734124$ |
Torsion generators
\( \left(1596, -798\right) \), \( \left(-560, 16450\right) \)
Integral points
\( \left(-1220, 610\right) \), \( \left(-1148, 12922\right) \), \( \left(-1148, -11774\right) \), \( \left(-980, 19810\right) \), \( \left(-980, -18830\right) \), \( \left(-770, 20860\right) \), \( \left(-770, -20090\right) \), \( \left(-560, 16450\right) \), \( \left(-560, -15890\right) \), \( \left(-428, 9322\right) \), \( \left(-428, -8894\right) \), \( \left(-420, 8610\right) \), \( \left(-420, -8190\right) \), \( \left(1596, -798\right) \), \( \left(1750, 30310\right) \), \( \left(1750, -32060\right) \), \( \left(1780, 33610\right) \), \( \left(1780, -35390\right) \), \( \left(2380, 87010\right) \), \( \left(2380, -89390\right) \), \( \left(3850, 217840\right) \), \( \left(3850, -221690\right) \), \( \left(4060, 238210\right) \), \( \left(4060, -242270\right) \), \( \left(7996, 698722\right) \), \( \left(7996, -706718\right) \), \( \left(8680, 792610\right) \), \( \left(8680, -801290\right) \), \( \left(31780, 5643610\right) \), \( \left(31780, -5675390\right) \), \( \left(37660, 7284130\right) \), \( \left(37660, -7321790\right) \), \( \left(4790380, 10482279010\right) \), \( \left(4790380, -10487069390\right) \)
Invariants
Conductor: | \( 53130 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $351347432901696000000 $ | = | $2^{12} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{2} \cdot 23^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{1007588745830352584072161}{351347432901696000000} \) | = | $2^{-12} \cdot 3^{-6} \cdot 5^{-6} \cdot 7^{-6} \cdot 11^{-2} \cdot 13^{6} \cdot 23^{-2} \cdot 593209^{3}$ | 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: | $2.6438695725897133721292986067\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $2.6438695725897133721292986067\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1.3837175901661311603814734124\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.12917284477344934694989264382\dots$ | 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: | $ 10368 $ = $ ( 2^{2} \cdot 3 )\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2\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: | $12$ | 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) $ ≈ $ 12.869189098907116387720923574 $ | 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
$\displaystyle 12.869189099 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.129173 \cdot 1.383718 \cdot 10368}{12^2} \approx 12.869189099$
Modular invariants
Modular form 53130.2.a.cs
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) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$3$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$5$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$7$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$11$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$23$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
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.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 106260 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 23 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 27727 & 6 \\ 36954 & 106255 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 60714 & 106255 \end{array}\right),\left(\begin{array}{rr} 57961 & 12 \\ 28986 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 106244 & 106253 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 53137 & 12 \\ 106194 & 106147 \end{array}\right),\left(\begin{array}{rr} 35421 & 10 \\ 35428 & 81 \end{array}\right),\left(\begin{array}{rr} 63757 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 106249 & 12 \\ 106248 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[106260])$ is a degree-$40951647830016000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/106260\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 53130.cs
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{6}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$4$ | \(\Q(\sqrt{-11}, \sqrt{-15})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{15}, \sqrt{-161})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{11}, \sqrt{161})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$6$ | 6.0.110623106187.1 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$9$ | 9.3.6916544024049336067083000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | 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 | split | split | split | nonsplit | ord | ord | ord | nonsplit | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 5 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | 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.