Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-239x+1181\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-239xz^2+1181z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3819x+71782\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 149454 \) | = | $2 \cdot 3^{2} \cdot 19^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $326855898$ | = | $2 \cdot 3^{9} \cdot 19^{2} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( \frac{5714497}{1242} \) | = | $2^{-1} \cdot 3^{-3} \cdot 19 \cdot 23^{-1} \cdot 67^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.34759149118742516346820244661$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.69245448300770309223092474383$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7929850986864758$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.353308529946894$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $1.6182493339419784005525475414$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $6.4729973357679136022101901657 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 6.472997336 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.618249 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 6.472997336\end{aligned}$$
Modular invariants
Modular form 149454.2.a.cl
For more coefficients, see the Downloads section to the right.
| Modular degree: | 69120 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $19$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $23$ | $1$ | $I_{1}$ | 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 \( 552 = 2^{3} \cdot 3 \cdot 23 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 551 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 2 \\ 97 & 3 \end{array}\right),\left(\begin{array}{rr} 277 & 2 \\ 277 & 3 \end{array}\right),\left(\begin{array}{rr} 415 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 551 & 2 \\ 550 & 3 \end{array}\right),\left(\begin{array}{rr} 185 & 2 \\ 185 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[552])$ is a degree-$9848881152$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/552\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 74727 = 3^{2} \cdot 19^{2} \cdot 23 \) |
| $3$ | additive | $2$ | \( 16606 = 2 \cdot 19^{2} \cdot 23 \) |
| $19$ | additive | $74$ | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 149454u consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 49818r1, its twist by $-3$.
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 |
|---|---|---|---|
| $3$ | 3.3.199272.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.21919550151168.1 | \(\Z/2\Z \oplus \Z/2\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | add | ord | ord | ord | ss | ord | add | nonsplit | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 8 | - | 0 | 0 | 0 | 0,0 | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.