Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2109000x+1178862500\)
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(homogenize, simplify) |
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\(y^2z=x^3-2109000xz^2+1178862500z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2109000x+1178862500\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(700, 6750)$ | $1.3361079606270161082661682943$ | $\infty$ |
Integral points
\((700,\pm 6750)\)
Invariants
| Conductor: | $N$ | = | \( 128700 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-469111500000000$ | = | $-1 \cdot 2^{8} \cdot 3^{8} \cdot 5^{9} \cdot 11 \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{2846137769984}{1287} \) | = | $-1 \cdot 2^{13} \cdot 3^{-2} \cdot 11^{-1} \cdot 13^{-1} \cdot 19^{3} \cdot 37^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1548731729981345460048155529$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.063609526034792453588197979784$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9296410090531607$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.700182915330119$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3361079606270161082661682943$ |
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| Real period: | $\Omega$ | ≈ | $0.42902296195238164578583724914$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 3\cdot2\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.8786519370763027902307720168 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.878651937 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.429023 \cdot 1.336108 \cdot 12}{1^2} \\ & \approx 6.878651937\end{aligned}$$
Modular invariants
Modular form 128700.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1489920 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | 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 \( 1430 = 2 \cdot 5 \cdot 11 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 287 & 2 \\ 287 & 3 \end{array}\right),\left(\begin{array}{rr} 1429 & 2 \\ 1428 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1429 & 0 \end{array}\right),\left(\begin{array}{rr} 651 & 2 \\ 651 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1211 & 2 \\ 1211 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[1430])$ is a degree-$498161664000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1430\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 6435 = 3^{2} \cdot 5 \cdot 11 \cdot 13 \) |
| $3$ | additive | $8$ | \( 14300 = 2^{2} \cdot 5^{2} \cdot 11 \cdot 13 \) |
| $5$ | additive | $14$ | \( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \) |
| $11$ | split multiplicative | $12$ | \( 11700 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 128700.q consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 42900.bl1, its twist by $-15$.
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 |
|---|---|---|---|
| $3$ | 3.1.2860.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.5848414000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | 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 | add | add | add | ord | split | nonsplit | ord | ord | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | - | 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 |
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.