Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-488x+5788\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-488xz^2+5788z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-631827x+271952046\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-26, 45)$ | $0.97792258275219982224316625958$ | $\infty$ |
| $(4, 60)$ | $0$ | $3$ |
Integral points
\( \left(-26, 45\right) \), \( \left(-26, -20\right) \), \( \left(4, 60\right) \), \( \left(4, -65\right) \), \( \left(604, 14535\right) \), \( \left(604, -15140\right) \)
Invariants
| Conductor: | $N$ | = | \( 1430 \) | = | $2 \cdot 5 \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-7261718750$ | = | $-1 \cdot 2 \cdot 5^{9} \cdot 11 \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{12814546750201}{7261718750} \) | = | $-1 \cdot 2^{-1} \cdot 5^{-9} \cdot 7^{3} \cdot 11^{-1} \cdot 13^{-2} \cdot 3343^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.59526072814650177826908306231$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.59526072814650177826908306231$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9116150928853675$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.248104475166794$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.97792258275219982224316625958$ |
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| Real period: | $\Omega$ | ≈ | $1.2284633721535502938335764565$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 18 $ = $ 1\cdot3^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.4026841474257534679574617951 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.402684147 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.228463 \cdot 0.977923 \cdot 18}{3^2} \\ & \approx 2.402684147\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 864 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 9.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 51480 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 18 \\ 47070 & 50857 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 12871 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 42121 & 18 \\ 18729 & 163 \end{array}\right),\left(\begin{array}{rr} 12881 & 25758 \\ 25578 & 18325 \end{array}\right),\left(\begin{array}{rr} 25741 & 18 \\ 25749 & 163 \end{array}\right),\left(\begin{array}{rr} 10297 & 18 \\ 41193 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 51463 & 18 \\ 51462 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[51480])$ is a degree-$6886586843136000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/51480\Z)$.
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 | $4$ | \( 55 = 5 \cdot 11 \) |
| $3$ | good | $2$ | \( 286 = 2 \cdot 11 \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 286 = 2 \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 130 = 2 \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 110 = 2 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 1430.c
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 |
|---|---|---|---|
| $3$ | 3.1.440.1 | \(\Z/6\Z\) | not in database |
| $3$ | 3.3.169.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.85184000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.180645811632.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.1068910128.4 | \(\Z/9\Z\) | not in database |
| $9$ | 9.3.411166897856000.2 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.5894998379219352129392827507027968.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.182603469794698651560072224857698336768000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.37831094993545145780591737700352000000.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.14401055232956963524803559424000000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | nonsplit | ord | split | ord | nonsplit | split | ord | ord | ss | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 3 | 1 | 2 | 1 | 1 | 2 | 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 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.