Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-135068x+34864607\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-135068xz^2+34864607z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2161083x+2229173782\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-267, 7333\right) \) | $0.69787154276763280605065794307$ | $\infty$ |
| \( \left(-459, 229\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-267:7333:1]\) | $0.69787154276763280605065794307$ | $\infty$ |
| \([-459:229:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-1069, 57600\right) \) | $0.69787154276763280605065794307$ | $\infty$ |
| \( \left(-1837, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-459, 229\right) \), \( \left(-267, 7333\right) \), \( \left(-267, -7067\right) \), \( \left(27, 5575\right) \), \( \left(27, -5603\right) \), \( \left(565, 11493\right) \), \( \left(565, -12059\right) \), \( \left(2133, 96133\right) \), \( \left(2133, -98267\right) \)
\([-459:229:1]\), \([-267:7333:1]\), \([-267:-7067:1]\), \([27:5575:1]\), \([27:-5603:1]\), \([565:11493:1]\), \([565:-12059:1]\), \([2133:96133:1]\), \([2133:-98267:1]\)
\( \left(-1837, 0\right) \), \((-1069,\pm 57600)\), \((107,\pm 44712)\), \((2259,\pm 94208)\), \((8531,\pm 777600)\)
Invariants
| Conductor: | $N$ | = | \( 23670 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 263$ |
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| Minimal Discriminant: | $\Delta$ | = | $-366396002795520000$ | = | $-1 \cdot 2^{22} \cdot 3^{12} \cdot 5^{4} \cdot 263 $ |
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| j-invariant: | $j$ | = | \( -\frac{373809708740405881}{502600826880000} \) | = | $-1 \cdot 2^{-22} \cdot 3^{-6} \cdot 5^{-4} \cdot 263^{-1} \cdot 720361^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0629004204185052740124013783$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5135942760844504283147787598$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9623698007613598$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.791039215492192$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.69787154276763280605065794307$ |
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| Real period: | $\Omega$ | ≈ | $0.27228223224732466514357354811$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 176 $ = $ ( 2 \cdot 11 )\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.3607929454128359903097243970 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.360792945 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.272282 \cdot 0.697872 \cdot 176}{2^2} \\ & \approx 8.360792945\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 337920 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $22$ | $I_{22}$ | split multiplicative | -1 | 1 | 22 | 22 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $263$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6312 = 2^{3} \cdot 3 \cdot 263 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6309 & 4 \\ 6308 & 5 \end{array}\right),\left(\begin{array}{rr} 3157 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 2105 & 4 \\ 4210 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 793 & 5524 \\ 3944 & 2367 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 2114 & 1 \\ 5255 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[6312])$ is a degree-$29282858237952$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6312\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 2367 = 3^{2} \cdot 263 \) |
| $3$ | additive | $6$ | \( 2630 = 2 \cdot 5 \cdot 263 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 4734 = 2 \cdot 3^{2} \cdot 263 \) |
| $11$ | good | $2$ | \( 11835 = 3^{2} \cdot 5 \cdot 263 \) |
| $263$ | nonsplit multiplicative | $264$ | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 23670.p
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 7890.e2, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-263}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{22 +16 \sqrt{6}})\) | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1587332691726336.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 263 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | add | nonsplit | ord | ss | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | 3 | - | 3 | 3 | 1,1 | 1 | 1 | 1,3 | 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 | 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.