Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+6132x-309680\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+6132xz^2-309680z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+7947045x-14472271242\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(114, 1312\right)\) |
$\hat{h}(P)$ | ≈ | $0.93902319515132906059103880344$ |
Torsion generators
\( \left(40, -20\right) \)
Integral points
\( \left(40, -20\right) \), \( \left(44, 192\right) \), \( \left(44, -236\right) \), \( \left(114, 1312\right) \), \( \left(114, -1426\right) \), \( \left(188, 2644\right) \), \( \left(188, -2832\right) \), \( \left(2778, 145094\right) \), \( \left(2778, -147872\right) \)
Invariants
Conductor: | \( 19166 \) | = | $2 \cdot 7 \cdot 37^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-56322826130368 $ | = | $-1 \cdot 2^{6} \cdot 7^{3} \cdot 37^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{9938375}{21952} \) | = | $2^{-6} \cdot 5^{3} \cdot 7^{-3} \cdot 43^{3}$ | 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: | $1.3226802021311691793585915112\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $-0.48277875419094304282545632432\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $0.9869508090989833\dots$ | |||
Szpiro ratio: | $3.9360564962994595\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.93902319515132906059103880344\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.32573061108301131601708589796\dots$ | 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: | $ 72 $ = $ ( 2 \cdot 3 )\cdot3\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'(E,1) $ ≈ $ 5.5056347851997556642322042116 $ | 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
$\displaystyle 5.505634785 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.325731 \cdot 0.939023 \cdot 72}{2^2} \approx 5.505634785$
Modular invariants
Modular form 19166.2.a.a
For more coefficients, see the Downloads section to the right.
Modular degree: | 51840 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$7$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$37$ | $4$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
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 | 8.6.0.1 |
$3$ | 3Cs | 3.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 18648 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 37 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 2110 & 12099 \\ 10989 & 3220 \end{array}\right),\left(\begin{array}{rr} 1333 & 12876 \\ 16650 & 667 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 1440 & 1819 \end{array}\right),\left(\begin{array}{rr} 18613 & 36 \\ 18612 & 37 \end{array}\right),\left(\begin{array}{rr} 9325 & 14652 \\ 16650 & 2665 \end{array}\right),\left(\begin{array}{rr} 9325 & 14652 \\ 0 & 1037 \end{array}\right),\left(\begin{array}{rr} 4031 & 0 \\ 0 & 18647 \end{array}\right)$.
The torsion field $K:=\Q(E[18648])$ is a degree-$25391279112192$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/18648\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 19166.a
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14.a6, its twist by $37$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{37}) \) | \(\Z/6\Z\) | 2.2.37.1-196.1-h3 |
$2$ | \(\Q(\sqrt{-111}) \) | \(\Z/6\Z\) | Not in database |
$4$ | 4.2.613312.6 | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{37})\) | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-7}, \sqrt{37})\) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-7}, \sqrt{-111})\) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$8$ | 8.0.225785003508736.23 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.18431428857856.28 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.364488705441.8 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$8$ | 8.4.376151609344.1 | \(\Z/12\Z\) | Not in database |
$8$ | 8.0.30468280356864.103 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/3\Z \oplus \Z/12\Z\) | 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 |
$16$ | 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 |
$18$ | 18.6.696906606797966288843602962074253.1 | \(\Z/18\Z\) | Not in database |
$18$ | 18.0.56185703389611501417712177571870809288704.2 | \(\Z/18\Z\) | 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 | split | ord | ss | split | ss | ord | ord | ord | ss | ord | ord | add | ord | ord | ord |
$\lambda$-invariant(s) | 2 | 7 | 1,3 | 2 | 1,1 | 1 | 1 | 3 | 1,1 | 1 | 1 | - | 1 | 3 | 1 |
$\mu$-invariant(s) | 0 | 1 | 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.