Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-49423080x+130545230400\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-49423080xz^2+130545230400z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-64052311707x+6090910426477494\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{8}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(3480, 24720\right)\) |
$\hat{h}(P)$ | ≈ | $2.0348652421815936965216376253$ |
Torsion generators
\( \left(3536, -1768\right) \), \( \left(3060, 87720\right) \)
Integral points
\( \left(-7440, 297720\right) \), \( \left(-7440, -290280\right) \), \( \left(-4080, 516120\right) \), \( \left(-4080, -512040\right) \), \( \left(-1938, 468996\right) \), \( \left(-1938, -467058\right) \), \( \left(-510, 394740\right) \), \( \left(-510, -394230\right) \), \( \left(960, 289320\right) \), \( \left(960, -290280\right) \), \( \left(3060, 87720\right) \), \( \left(3060, -90780\right) \), \( \left(3480, 24720\right) \), \( \left(3480, -28200\right) \), \( \left(3536, -1768\right) \), \( \left(4560, -2280\right) \), \( \left(4760, 53720\right) \), \( \left(4760, -58480\right) \), \( \left(5100, 103020\right) \), \( \left(5100, -108120\right) \), \( \left(5712, 183192\right) \), \( \left(5712, -188904\right) \), \( \left(6480, 283800\right) \), \( \left(6480, -290280\right) \), \( \left(8160, 516120\right) \), \( \left(8160, -524280\right) \), \( \left(11310, 1003470\right) \), \( \left(11310, -1014780\right) \), \( \left(15810, 1808970\right) \), \( \left(15810, -1824780\right) \), \( \left(28560, 4677720\right) \), \( \left(28560, -4706280\right) \), \( \left(69360, 18141720\right) \), \( \left(69360, -18211080\right) \), \( \left(216240, 100394520\right) \), \( \left(216240, -100610760\right) \)
Invariants
Conductor: | \( 82110 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $363628103290905600000000 $ | = | $2^{16} \cdot 3^{8} \cdot 5^{8} \cdot 7^{2} \cdot 17^{4} \cdot 23^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{13350979617415439280823624321}{363628103290905600000000} \) | = | $2^{-16} \cdot 3^{-8} \cdot 5^{-8} \cdot 7^{-2} \cdot 17^{-4} \cdot 23^{-2} \cdot 241^{3} \cdot 9843601^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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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: | $3.3009530286292500950151736916\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $3.3009530286292500950151736916\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $0.9923666812811367\dots$ | |||
Szpiro ratio: | $5.723086384275005\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $2.0348652421815936965216376253\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.095224619621214896073847388293\dots$ | 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: | $ 16384 $ = $ 2^{4}\cdot2^{3}\cdot2^{3}\cdot2\cdot2^{2}\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: | $16$ | 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);
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Special value: | $ L'(E,1) $ ≈ $ 12.401233194699109664652645957 $ | 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
$\displaystyle 12.401233195 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.095225 \cdot 2.034865 \cdot 16384}{16^2} \approx 12.401233195$
Modular invariants
Modular form 82110.2.a.bs
For more coefficients, see the Downloads section to the right.
Modular degree: | 14680064 | 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 $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
$3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
$5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
$7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
$17$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
$23$ | $2$ | $I_{2}$ | nonsplit 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 | 8.96.0.40 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 656880 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 199921 & 16 \\ 542646 & 97 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 74 & 574849 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 618249 & 16 \\ 68 & 121 \end{array}\right),\left(\begin{array}{rr} 437921 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 469209 & 16 \\ 187874 & 345 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 4 & 492673 \end{array}\right),\left(\begin{array}{rr} 131377 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 656865 & 16 \\ 656864 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[656880])$ is a degree-$31107765182178263040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/656880\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | split multiplicative | $4$ | \( 1 \) |
$3$ | split multiplicative | $4$ | \( 27370 = 2 \cdot 5 \cdot 7 \cdot 17 \cdot 23 \) |
$5$ | split multiplicative | $6$ | \( 16422 = 2 \cdot 3 \cdot 7 \cdot 17 \cdot 23 \) |
$7$ | nonsplit multiplicative | $8$ | \( 11730 = 2 \cdot 3 \cdot 5 \cdot 17 \cdot 23 \) |
$17$ | split multiplicative | $18$ | \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \) |
$23$ | nonsplit multiplicative | $24$ | \( 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 82110bt
consists of 8 curves linked by isogenies of
degrees dividing 16.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{8}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$4$ | \(\Q(i, \sqrt{161})\) | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$4$ | \(\Q(\sqrt{-15}, \sqrt{-119})\) | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
$4$ | \(\Q(\sqrt{15}, \sqrt{391})\) | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
$16$ | deg 16 | \(\Z/8\Z \oplus \Z/8\Z\) | not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/32\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/32\Z\) | 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 | nonsplit | ord | ord | split | ord | nonsplit | ord | ss | ord | ord | ord | ss |
$\lambda$-invariant(s) | 10 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | 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.