Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-4035x+11842\)
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(homogenize, simplify) |
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\(y^2z=x^3-4035xz^2+11842z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4035x+11842\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1, 126)$ | $0.92645395941797198713339222599$ | $\infty$ |
| $(62, 0)$ | $0$ | $2$ |
Integral points
\((-57,\pm 238)\), \((-1,\pm 126)\), \( \left(62, 0\right) \), \((258,\pm 4018)\)
Invariants
| Conductor: | $N$ | = | \( 17136 \) | = | $2^{4} \cdot 3^{2} \cdot 7 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $4143882903552$ | = | $2^{13} \cdot 3^{6} \cdot 7^{4} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{2433138625}{1387778} \) | = | $2^{-1} \cdot 5^{3} \cdot 7^{-4} \cdot 17^{-2} \cdot 269^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1108499378682980611819982041$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.13160338702570209393285653582$ |
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| $abc$ quality: | $Q$ | ≈ | $0.962208541820214$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7462431597905232$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.92645395941797198713339222599$ |
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| Real period: | $\Omega$ | ≈ | $0.66911351797608086649375303204$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.9592229442322279423485218012 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.959222944 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.669114 \cdot 0.926454 \cdot 32}{2^2} \\ & \approx 4.959222944\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18432 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $17$ | $2$ | $I_{2}$ | split 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 |
|---|---|---|
| $2$ | 2B | 8.6.0.6 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 136 = 2^{3} \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 133 & 4 \\ 132 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 67 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 121 & 18 \\ 16 & 119 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 105 & 4 \\ 74 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[136])$ is a degree-$10027008$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/136\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 9 = 3^{2} \) |
| $3$ | additive | $6$ | \( 1904 = 2^{4} \cdot 7 \cdot 17 \) |
| $7$ | split multiplicative | $8$ | \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 17136bm
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 238d2, its twist by $12$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.4.20808.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.392737849344.39 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.27710263296.1 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | add | add | ss | split | ord | ord | split | ss | ord | ord | ss | ord | ord | ss | ss |
| $\lambda$-invariant(s) | - | - | 1,1 | 2 | 1 | 1 | 2 | 1,3 | 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 | 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.