Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-283467x+57972069\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-283467xz^2+57972069z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-367373907x+2710255456494\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(305, -93)$ | $0.53019199694539033989525655543$ | $\infty$ |
| $(1227/4, -1227/8)$ | $0$ | $2$ |
Integral points
\( \left(273, 876\right) \), \( \left(273, -1149\right) \), \( \left(305, -93\right) \), \( \left(305, -212\right) \), \( \left(319, 208\right) \), \( \left(319, -527\right) \), \( \left(543, 7761\right) \), \( \left(543, -8304\right) \)
Invariants
| Conductor: | $N$ | = | \( 82110 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $2326887372600$ | = | $2^{3} \cdot 3^{6} \cdot 5^{2} \cdot 7^{4} \cdot 17^{2} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( \frac{2519031674995639224121}{2326887372600} \) | = | $2^{-3} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{-4} \cdot 17^{-2} \cdot 23^{-1} \cdot 107^{3} \cdot 127163^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6700356842906770192312572861$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6700356842906770192312572861$ |
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| $abc$ quality: | $Q$ | ≈ | $0.940764964415334$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.35480441573195$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.53019199694539033989525655543$ |
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| Real period: | $\Omega$ | ≈ | $0.68503328581425353510567943385$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2\cdot2\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.9055933262393713435459026429 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.905593326 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.685033 \cdot 0.530192 \cdot 32}{2^2} \\ & \approx 2.905593326\end{aligned}$$
Modular invariants
Modular form 82110.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 718848 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $17$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $23$ | $1$ | $I_{1}$ | nonsplit 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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 2402 & 1 \\ 1559 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1841 & 4 \\ 922 & 9 \end{array}\right),\left(\begin{array}{rr} 2757 & 4 \\ 2756 & 5 \end{array}\right),\left(\begin{array}{rr} 1657 & 4 \\ 554 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 1379 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1036 & 1729 \\ 345 & 2416 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[2760])$ is a degree-$787910492160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2760\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 23 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 13685 = 5 \cdot 7 \cdot 17 \cdot 23 \) |
| $5$ | split multiplicative | $6$ | \( 16422 = 2 \cdot 3 \cdot 7 \cdot 17 \cdot 23 \) |
| $7$ | split multiplicative | $8$ | \( 11730 = 2 \cdot 3 \cdot 5 \cdot 17 \cdot 23 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \) |
| $23$ | 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.
Its isogeny class 82110.m
consists of 2 curves linked by isogenies of
degree 2.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{46}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.165600.1 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.928445276160000.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | split | split | ord | ord | nonsplit | ss | nonsplit | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 8 | 1 | 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.