Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-450807x-236547235\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-450807xz^2-236547235z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7212915x-15146235954\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5160816539/376996, 368689126957875/231475544)$ | $19.260956678104592162310317160$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 85698 \) | = | $2 \cdot 3^{4} \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-18331984114873270272$ | = | $-1 \cdot 2^{21} \cdot 3^{10} \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{1159088625}{2097152} \) | = | $-1 \cdot 2^{-21} \cdot 3^{2} \cdot 5^{3} \cdot 101^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3864213311762698729318392851$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.096836017345063048634241494907$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1123490200903752$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.585731075767879$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $19.260956678104592162310317160$ |
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| Real period: | $\Omega$ | ≈ | $0.086882597649458473979514919320$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.3468838988148230724694425228 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.346883899 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.086883 \cdot 19.260957 \cdot 2}{1^2} \\ & \approx 3.346883899\end{aligned}$$
Modular invariants
Modular form 85698.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1496880 |
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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 not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{21}$ | nonsplit multiplicative | 1 | 1 | 21 | 21 |
| $3$ | $1$ | $IV^{*}$ | additive | -1 | 4 | 10 | 0 |
| $23$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2G | 8.2.0.1 |
| $3$ | 3B | 3.4.0.1 |
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11592 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 23 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 967 & 1794 \\ 1932 & 4831 \end{array}\right),\left(\begin{array}{rr} 1 & 3450 \\ 10626 & 5797 \end{array}\right),\left(\begin{array}{rr} 6807 & 782 \\ 2576 & 5703 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 7728 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1932 & 1 \end{array}\right),\left(\begin{array}{rr} 9661 & 10626 \\ 7245 & 10627 \end{array}\right),\left(\begin{array}{rr} 1007 & 0 \\ 0 & 11591 \end{array}\right),\left(\begin{array}{rr} 1 & 8280 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4831 & 10626 \\ 6279 & 6763 \end{array}\right),\left(\begin{array}{rr} 11110 & 5865 \\ 7245 & 7729 \end{array}\right),\left(\begin{array}{rr} 3865 & 7728 \\ 3864 & 7729 \end{array}\right),\left(\begin{array}{rr} 1 & 7728 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1289 & 6440 \\ 5152 & 6441 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 966 & 1 \end{array}\right),\left(\begin{array}{rr} 6049 & 5544 \\ 6048 & 6049 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 5544 & 1 \end{array}\right),\left(\begin{array}{rr} 6625 & 8970 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[11592])$ is a degree-$4188236709888$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11592\Z)$.
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 | $4$ | \( 42849 = 3^{4} \cdot 23^{2} \) |
| $3$ | additive | $4$ | \( 23 \) |
| $7$ | good | $2$ | \( 42849 = 3^{4} \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 162 = 2 \cdot 3^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 85698.e
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
The minimal quadratic twist of this elliptic curve is 162.b2, its twist by $-23$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{69}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.648.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.718449183.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.4024991806227.4 | \(\Z/7\Z\) | not in database |
| $6$ | 6.2.15326915904.3 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/21\Z\) | not in database |
| $18$ | 18.6.29897811966303126825577195514351616.4 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.637819988614466705612313504306167808.2 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.17093654581891449594915499403756652221693952.1 | \(\Z/14\Z\) | 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 | add | ss | ord | ord | ord | ord | ord | add | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 8 | - | 1,1 | 1 | 1 | 1 | 1 | 1 | - | 1 | 3 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | - | 0,0 | 1 | 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.