Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2+7482x-250927\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z+7482xz^2-250927z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+9696240x-11823595056\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5853/196, 12095/2744)$ | $4.0888418962203437729030361407$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 13467 \) | = | $3 \cdot 67^{2}$ |
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| Discriminant: | $\Delta$ | = | $-54546404447907$ | = | $-1 \cdot 3^{2} \cdot 67^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{512000}{603} \) | = | $2^{12} \cdot 3^{-2} \cdot 5^{3} \cdot 67^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3221126799603381250813227836$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.78023362973514490475371321458$ |
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| $abc$ quality: | $Q$ | ≈ | $0.800359374985248$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.039626398918516$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.0888418962203437729030361407$ |
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| Real period: | $\Omega$ | ≈ | $0.33820227810273563003318618836$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.062845152829036818025649677 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.062845153 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.338202 \cdot 4.088842 \cdot 8}{1^2} \\ & \approx 11.062845153\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 53856 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $67$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 | $2$ | \( 4489 = 67^{2} \) |
| $3$ | split multiplicative | $4$ | \( 4489 = 67^{2} \) |
| $67$ | additive | $2312$ | \( 3 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 13467k consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 201a1, its twist by $-67$.
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 |
|---|---|---|---|
| $3$ | 3.1.268.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.4812208.1 | \(\Z/2\Z \oplus \Z/2\Z\) | 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 | ss | split | ss | ss | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord | ord | add |
| $\lambda$-invariant(s) | 2,3 | 2 | 3,1 | 1,3 | 3 | 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 | 0,0 | 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.