Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-1871137x-776514847\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-1871137xz^2-776514847z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2424992931x-36221801711202\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-5929073609/5616900, 70475306031929/13312053000)$ | $17.693935795819770857268433077$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 123981 \) | = | $3 \cdot 11 \cdot 13 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $158891791677930793911$ | = | $3 \cdot 11^{2} \cdot 13^{7} \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{103860107394697}{22777711671} \) | = | $3^{-1} \cdot 11^{-2} \cdot 13^{-7} \cdot 17 \cdot 101^{3} \cdot 181^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5898234948082400296925270914$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.70101459877076264285950401282$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9142722408575247$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.684542802577433$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $17.693935795819770857268433077$ |
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| Real period: | $\Omega$ | ≈ | $0.13110027515732359207484018924$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 1\cdot2\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $13.918079108687936082989038587 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.918079109 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.131100 \cdot 17.693936 \cdot 6}{1^2} \\ & \approx 13.918079109\end{aligned}$$
Modular invariants
Modular form 123981.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3975552 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $17$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
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 \( 156 = 2^{2} \cdot 3 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 155 & 2 \\ 154 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 53 & 2 \\ 53 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 155 & 0 \end{array}\right),\left(\begin{array}{rr} 145 & 2 \\ 145 & 3 \end{array}\right),\left(\begin{array}{rr} 79 & 2 \\ 79 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[156])$ is a degree-$60383232$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/156\Z)$.
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 | $2$ | \( 11271 = 3 \cdot 13 \cdot 17^{2} \) |
| $3$ | split multiplicative | $4$ | \( 41327 = 11 \cdot 13 \cdot 17^{2} \) |
| $7$ | good | $2$ | \( 9537 = 3 \cdot 11 \cdot 17^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 11271 = 3 \cdot 13 \cdot 17^{2} \) |
| $13$ | nonsplit multiplicative | $14$ | \( 9537 = 3 \cdot 11 \cdot 17^{2} \) |
| $17$ | additive | $114$ | \( 429 = 3 \cdot 11 \cdot 13 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 123981n consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 123981d1, its twist by $17$.
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 |
|---|---|---|---|
| $3$ | 3.3.45084.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.317080460736.1 | \(\Z/2\Z \oplus \Z/2\Z\) | 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 | ord | split | ord | ord | nonsplit | nonsplit | add | ord | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 4 | 1 | 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 |
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.