Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-319297x-93143361\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-319297xz^2-93143361z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-25863084x-67823920944\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 118976 \) | = | $2^{6} \cdot 11 \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1655968550990381056$ | = | $-1 \cdot 2^{24} \cdot 11^{2} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{16835377}{7744} \) | = | $-1 \cdot 2^{-6} \cdot 11^{-2} \cdot 13 \cdot 109^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2022992798634568605262045609$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.54738772928415226096863524900$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8587093944863271$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.297214626539419$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $0.098244112410156688236072401556$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.39297644964062675294428960622 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.392976450 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.098244 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 0.392976450\end{aligned}$$
Modular invariants
Modular form 118976.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1916928 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{14}^{*}$ | additive | -1 | 6 | 24 | 6 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 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 | $2$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $32$, genus $0$, and generators
$\left(\begin{array}{rr} 131 & 0 \\ 0 & 263 \end{array}\right),\left(\begin{array}{rr} 232 & 9 \\ 69 & 28 \end{array}\right),\left(\begin{array}{rr} 123 & 259 \\ 257 & 260 \end{array}\right),\left(\begin{array}{rr} 253 & 12 \\ 252 & 13 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 226 & 249 \end{array}\right),\left(\begin{array}{rr} 259 & 262 \\ 170 & 147 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 211 & 256 \\ 192 & 239 \end{array}\right),\left(\begin{array}{rr} 143 & 252 \\ 144 & 251 \end{array}\right),\left(\begin{array}{rr} 4 & 9 \\ 3 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 119 & 255 \\ 225 & 236 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$30412800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 | $74$ | \( 704 = 2^{6} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 118976cl
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 3718k1, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 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 |
| $6$ | 6.2.5780665972224.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.58492928.3 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.13685690504052736.18 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.31273687642709647835379659333173248.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.12362707640976622756860426992018717147136.2 | \(\Z/6\Z\) | 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 | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | 0 | 0 | 0 | - | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-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
All $p$-adic regulators are identically $1$ since the rank is $0$.