Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-685908x-228577454\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-685908xz^2-228577454z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-888936147x-10661842873746\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(961, -481)$ | $0$ | $2$ |
Integral points
\( \left(961, -481\right) \)
Invariants
| Conductor: | $N$ | = | \( 10830 \) | = | $2 \cdot 3 \cdot 5 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1907006848826423040$ | = | $-1 \cdot 2^{8} \cdot 3^{5} \cdot 5 \cdot 19^{10} $ |
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| j-invariant: | $j$ | = | \( -\frac{758575480593601}{40535043840} \) | = | $-1 \cdot 2^{-8} \cdot 3^{-5} \cdot 5^{-1} \cdot 11^{3} \cdot 19^{-4} \cdot 8291^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2665606969436864439914138732$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.79434120736046621398690015726$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9830842927041407$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.599245947694625$ | |||
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: | $\Omega$ | ≈ | $0.082690495767993401669175329647$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 2\cdot5\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.3076198307197360667670131859 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.307619831 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.082690 \cdot 1.000000 \cdot 40}{2^2} \\ & \approx 3.307619831\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 345600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $19$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
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 |
|---|---|---|
| $2$ | 2B | 8.12.0.6 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 359 & 2272 \\ 1436 & 2247 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 859 & 858 \\ 298 & 1435 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1433 & 1428 \\ 1430 & 287 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2274 & 2275 \end{array}\right),\left(\begin{array}{rr} 2273 & 8 \\ 2272 & 9 \end{array}\right),\left(\begin{array}{rr} 1528 & 3 \\ 1525 & 2 \end{array}\right),\left(\begin{array}{rr} 1376 & 3 \\ 5 & 2 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$90773913600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
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 | $4$ | \( 5415 = 3 \cdot 5 \cdot 19^{2} \) |
| $3$ | split multiplicative | $4$ | \( 3610 = 2 \cdot 5 \cdot 19^{2} \) |
| $5$ | split multiplicative | $6$ | \( 722 = 2 \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 10830.p
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 570.i4, its twist by $-19$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{285}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-19}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-19})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.1218375.2 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.380016036000000.43 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7506489600.4 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.1484437640625.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | 19 |
|---|---|---|---|---|
| Reduction type | nonsplit | split | split | add |
| $\lambda$-invariant(s) | 2 | 1 | 3 | - |
| $\mu$-invariant(s) | 0 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
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 $0$.