Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-4964x-138063\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-4964xz^2-138063z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6432723x-6422157522\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(131, 1144)$ | $1.1452820142880711443491568689$ | $\infty$ |
| $(1371/4, 48235/8)$ | $4.1389053651612204674056395649$ | $\infty$ |
Integral points
\( \left(131, 1144\right) \), \( \left(131, -1276\right) \), \( \left(751, 20114\right) \), \( \left(751, -20866\right) \)
Invariants
| Conductor: | $N$ | = | \( 40535 \) | = | $5 \cdot 11^{2} \cdot 67$ |
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| Discriminant: | $\Delta$ | = | $-359051125675$ | = | $-1 \cdot 5^{2} \cdot 11^{8} \cdot 67 $ |
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| j-invariant: | $j$ | = | \( -\frac{63088729}{1675} \) | = | $-1 \cdot 5^{-2} \cdot 11 \cdot 67^{-1} \cdot 179^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.99955322473828437384533916891$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.59904362379396265552928988307$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7442005476351693$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.5050281295330343$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.6544954855378650143531333251$ |
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| Real period: | $\Omega$ | ≈ | $0.28394982786251769019837925819$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 2\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $7.9298591514320548079989669556 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.929859151 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.283950 \cdot 4.654495 \cdot 6}{1^2} \\ & \approx 7.929859151\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 76032 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $67$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 134 = 2 \cdot 67 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 133 & 0 \end{array}\right),\left(\begin{array}{rr} 133 & 2 \\ 132 & 3 \end{array}\right),\left(\begin{array}{rr} 69 & 2 \\ 69 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[134])$ is a degree-$59537808$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/134\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 8107 = 11^{2} \cdot 67 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 8107 = 11^{2} \cdot 67 \) |
| $11$ | additive | $52$ | \( 335 = 5 \cdot 67 \) |
| $67$ | nonsplit multiplicative | $68$ | \( 605 = 5 \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 40535b consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 40535d1, its twist by $-11$.
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 |
|---|---|---|---|
| $3$ | 3.1.8107.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.4403471083.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | 67 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ord | ord | nonsplit | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | 4 | 4 | 4 | 2 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.