Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-103987x+12897998\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-103987xz^2+12897998z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-134766531x+602173305918\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(184, -43)$ | $0$ | $3$ |
Integral points
\( \left(184, -43\right) \), \( \left(184, -142\right) \)
Invariants
| Conductor: | $N$ | = | \( 858 \) | = | $2 \cdot 3 \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-103332962304$ | = | $-1 \cdot 2^{13} \cdot 3^{6} \cdot 11^{3} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{124352595912593543977}{103332962304} \) | = | $-1 \cdot 2^{-13} \cdot 3^{-6} \cdot 11^{-3} \cdot 13^{-1} \cdot 17^{3} \cdot 47^{3} \cdot 6247^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4167896706078661440647732849$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4167896706078661440647732849$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0211200499526043$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.850091208039536$ | |||
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: | $\Omega$ | ≈ | $0.88470167261597405304185502254$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 18 $ = $ 1\cdot( 2 \cdot 3 )\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.7694033452319481060837100451 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.769403345 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.884702 \cdot 1.000000 \cdot 18}{3^2} \\ & \approx 1.769403345\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3120 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{13}$ | nonsplit multiplicative | 1 | 1 | 13 | 13 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $11$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 2641 & 6 \\ 1059 & 19 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 2718 & 721 \\ 3289 & 3147 \end{array}\right),\left(\begin{array}{rr} 1717 & 6 \\ 1719 & 19 \end{array}\right),\left(\begin{array}{rr} 937 & 6 \\ 2811 & 19 \end{array}\right),\left(\begin{array}{rr} 2575 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3427 & 6 \\ 3426 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3432])$ is a degree-$1594117324800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3432\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 143 = 11 \cdot 13 \) |
| $3$ | split multiplicative | $4$ | \( 26 = 2 \cdot 13 \) |
| $11$ | split multiplicative | $12$ | \( 78 = 2 \cdot 3 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 33 = 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 858d
consists of 2 curves linked by isogenies of
degree 3.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.1144.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.1497193984.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.12338352.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.11616313484322127152.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.9213703300212043070302000615784448.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 | 11 | 13 |
|---|---|---|---|---|
| Reduction type | nonsplit | split | split | split |
| $\lambda$-invariant(s) | 5 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.