Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-1545324081x+23381332267545\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-1545324081xz^2+23381332267545z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2002740009003x+1090885446494606502\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(22896, 37749\right) \) | $1.5655364685874058726718889498$ | $\infty$ |
| \( \left(\frac{91055}{4}, -\frac{91055}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([22896:37749:1]\) | $1.5655364685874058726718889498$ | $\infty$ |
| \([182110:-91055:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(824259, 10626552\right) \) | $1.5655364685874058726718889498$ | $\infty$ |
| \( \left(819498, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(22896, 37749\right) \), \( \left(22896, -60645\right) \), \( \left(92724, 25978851\right) \), \( \left(92724, -26071575\right) \)
\([22896:37749:1]\), \([22896:-60645:1]\), \([92724:25978851:1]\), \([92724:-26071575:1]\)
\((824259,\pm 10626552)\), \((3338067,\pm 5621446008)\)
Invariants
| Conductor: | $N$ | = | \( 82110 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23$ |
|
| Minimal Discriminant: | $\Delta$ | = | $6337051947007316223570000$ | = | $2^{4} \cdot 3^{5} \cdot 5^{4} \cdot 7 \cdot 17 \cdot 23^{12} $ |
|
| j-invariant: | $j$ | = | \( \frac{408114879566277798087624787060369}{6337051947007316223570000} \) | = | $2^{-4} \cdot 3^{-5} \cdot 5^{-4} \cdot 7^{-1} \cdot 17^{-1} \cdot 23^{-12} \cdot 74175555889^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8938380134538398702345354045$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.8938380134538398702345354045$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0162274694181177$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.635766039854349$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.5655364685874058726718889498$ |
|
| Real period: | $\Omega$ | ≈ | $0.068908088017700874294505230160$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 480 $ = $ 2^{2}\cdot5\cdot2\cdot1\cdot1\cdot( 2^{2} \cdot 3 ) $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $12.945374972680987655966475147 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 12.945374973 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.068908 \cdot 1.565536 \cdot 480}{2^2} \\ & \approx 12.945374973\end{aligned}$$
Modular invariants
Modular form 82110.2.a.br
For more coefficients, see the Downloads section to the right.
| Modular degree: | 42270720 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is semistable. There are 6 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $23$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 65688 = 2^{3} \cdot 3 \cdot 7 \cdot 17 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 41059 & 41058 \\ 57490 & 24643 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 38648 & 3 \\ 7733 & 2 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 65682 & 65683 \end{array}\right),\left(\begin{array}{rr} 65681 & 8 \\ 65680 & 9 \end{array}\right),\left(\begin{array}{rr} 2857 & 8 \\ 11428 & 33 \end{array}\right),\left(\begin{array}{rr} 8219 & 8212 \\ 8258 & 41061 \end{array}\right),\left(\begin{array}{rr} 37544 & 3 \\ 56309 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 43796 & 1 \\ 21919 & 6 \end{array}\right)$.
The torsion field $K:=\Q(E[65688])$ is a degree-$64807844129538048$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/65688\Z)$.
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 | $4$ | \( 357 = 3 \cdot 7 \cdot 17 \) |
| $3$ | split multiplicative | $4$ | \( 1190 = 2 \cdot 5 \cdot 7 \cdot 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 5474 = 2 \cdot 7 \cdot 17 \cdot 23 \) |
| $7$ | split multiplicative | $8$ | \( 11730 = 2 \cdot 3 \cdot 5 \cdot 17 \cdot 23 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \) |
| $23$ | split 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 and 4.
Its isogeny class 82110br
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{357}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{119}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{119})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | split | ord | ord | nonsplit | ord | split | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 8 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 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.