Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-4907046369x-132307275256961\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-4907046369xz^2-132307275256961z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-397470755916x-96450811250056848\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{4719000530314511102152877864748878268057989411}{49530914785242040009615929425148259275025}, \frac{179447282077503210399925610726274664630253782052960501998660693308416}{11023373522301146742328910021239775008993980308692248633562375}\right) \) | $100.51049109528268539634743029$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1050239143030999276148770534463929139628837772116087419172988225445:179447282077503210399925610726274664630253782052960501998660693308416:11023373522301146742328910021239775008993980308692248633562375]\) | $100.51049109528268539634743029$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{42471153365574955645495929630528179857299729774}{49530914785242040009615929425148259275025}, \frac{4845076616092586680797991489609415945016852115429933553963838719327232}{11023373522301146742328910021239775008993980308692248633562375}\right) \) | $100.51049109528268539634743029$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 118976 \) | = | $2^{6} \cdot 11 \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-4477738961877990375424$ | = | $-1 \cdot 2^{28} \cdot 11^{2} \cdot 13^{10} $ |
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| j-invariant: | $j$ | = | \( -\frac{361585288790756017}{123904} \) | = | $-1 \cdot 2^{-10} \cdot 11^{-2} \cdot 13^{2} \cdot 128857^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9488626182426360705691439687$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.77168404951810415973205625188$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0437212839117709$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.721792403986286$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $100.51049109528268539634743029$ |
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| Real period: | $\Omega$ | ≈ | $0.0090191273361063171850565459714$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.2521353424234771279006927067 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.252135342 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.009019 \cdot 100.510491 \cdot 8}{1^2} \\ & \approx 7.252135342\end{aligned}$$
Modular invariants
Modular form 118976.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 67092480 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{18}^{*}$ | additive | 1 | 6 | 28 | 10 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2G | 4.2.0.1 | $2$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1144 = 2^{3} \cdot 11 \cdot 13 \), index $4$, genus $0$, and generators
$\left(\begin{array}{rr} 571 & 0 \\ 0 & 1143 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 753 & 260 \\ 1040 & 77 \end{array}\right),\left(\begin{array}{rr} 791 & 0 \\ 0 & 1143 \end{array}\right),\left(\begin{array}{rr} 350 & 1053 \\ 1105 & 51 \end{array}\right),\left(\begin{array}{rr} 1141 & 4 \\ 1140 & 5 \end{array}\right),\left(\begin{array}{rr} 311 & 260 \\ 312 & 259 \end{array}\right),\left(\begin{array}{rr} 2 & 3 \\ 1139 & 1137 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 741 \\ 0 & 287 \end{array}\right)$.
The torsion field $K:=\Q(E[1144])$ is a degree-$132843110400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1144\Z)$.
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 | $2$ | \( 169 = 13^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 10816 = 2^{6} \cdot 13^{2} \) |
| $13$ | additive | $50$ | \( 704 = 2^{6} \cdot 11 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 118976.u consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 3718.s1, its twist by $104$.
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 |
|---|---|---|---|
| $3$ | 3.1.676.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1827904.2 | \(\Z/2\Z \oplus \Z/2\Z\) | 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 | ord | ord | ord | nonsplit | add | ord | ord | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 3 | 1 | 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 |
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.