Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-4268833x-3387702172\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-4268833xz^2-3387702172z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5532406947x-158040035304354\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1148, 32)$ | $2.6644758427199098480300336813$ | $\infty$ |
| $(-4953/4, 4949/8)$ | $0$ | $2$ |
Integral points
\( \left(-1148, 1115\right) \), \( \left(-1148, 32\right) \), \( \left(2422, 20750\right) \), \( \left(2422, -23173\right) \), \( \left(87268, 25729088\right) \), \( \left(87268, -25816357\right) \)
Invariants
| Conductor: | $N$ | = | \( 119130 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $21782957198729387760$ | = | $2^{4} \cdot 3^{3} \cdot 5 \cdot 11^{8} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{182864522286982801}{463015182960} \) | = | $2^{-4} \cdot 3^{-3} \cdot 5^{-1} \cdot 11^{-8} \cdot 567601^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5874055228761778260944541281$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1151860332929575960899404122$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0750149448569075$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.912243342772097$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.6644758427199098480300336813$ |
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| Real period: | $\Omega$ | ≈ | $0.10504813825388634673070928488$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2\cdot3\cdot1\cdot2^{3}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.7175574408043541453712925606 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.717557441 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.105048 \cdot 2.664476 \cdot 96}{2^2} \\ & \approx 6.717557441\end{aligned}$$
Modular invariants
Modular form 119130.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5308416 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $19$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 50160 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 19 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 42256 \\ 8588 & 24549 \end{array}\right),\left(\begin{array}{rr} 37298 & 31027 \\ 47215 & 30382 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 50156 & 50157 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 50062 & 50147 \end{array}\right),\left(\begin{array}{rr} 50145 & 16 \\ 50144 & 17 \end{array}\right),\left(\begin{array}{rr} 8456 & 2641 \\ 28063 & 23770 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 6176 & 13205 \\ 31635 & 10546 \end{array}\right),\left(\begin{array}{rr} 9121 & 42256 \\ 43928 & 37089 \end{array}\right),\left(\begin{array}{rr} 2639 & 0 \\ 0 & 50159 \end{array}\right)$.
The torsion field $K:=\Q(E[50160])$ is a degree-$4792862638080000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/50160\Z)$.
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 | $4$ | \( 39710 = 2 \cdot 5 \cdot 11 \cdot 19^{2} \) |
| $5$ | split multiplicative | $6$ | \( 23826 = 2 \cdot 3 \cdot 11 \cdot 19^{2} \) |
| $11$ | split multiplicative | $12$ | \( 10830 = 2 \cdot 3 \cdot 5 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 119130.s
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 330.d2, 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 |
| $4$ | \(\Q(\sqrt{5}, \sqrt{-19})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-19})\) | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.1688960160000.4 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | \(\Z/16\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | split | split | ss | split | ord | ord | add | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 2 | 2 | 1,1 | 2 | 1 | 1 | - | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 0 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.