Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-6231x-187253\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-6231xz^2-187253z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-504738x-138021624\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(217, 2938\right) \) | $3.9402194780871741153989346069$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([217:2938:1]\) | $3.9402194780871741153989346069$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1950, 79326\right) \) | $3.9402194780871741153989346069$ | $\infty$ |
Integral points
\((217,\pm 2938)\)
\([217:\pm 2938:1]\)
\((217,\pm 2938)\)
Invariants
| Conductor: | $N$ | = | \( 43264 \) | = | $2^{8} \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1124864$ | = | $-1 \cdot 2^{9} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( -23788477376 \) | = | $-1 \cdot 2^{6} \cdot 719^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.64641650939771798446279755761$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.51468121538762518161349839387$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0292750347112116$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.5433601935729717$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.9402194780871741153989346069$ |
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| Real period: | $\Omega$ | ≈ | $0.26867317775734181910252242684$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.2345251529562237589365133261 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.234525153 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.268673 \cdot 3.940219 \cdot 4}{1^2} \\ & \approx 4.234525153\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 19200 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 8 | 9 | 0 |
| $13$ | $2$ | $III$ | additive | -1 | 2 | 3 | 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 |
|---|---|---|---|
| $3$ | 3Nn | 9.9.0.1 | $9$ |
| $5$ | 5B | 5.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \), index $864$, genus $33$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 5760 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 450 \\ 0 & 2081 \end{array}\right),\left(\begin{array}{rr} 7381 & 3735 \\ 3375 & 1666 \end{array}\right),\left(\begin{array}{rr} 4681 & 4680 \\ 4680 & 4681 \end{array}\right),\left(\begin{array}{rr} 9271 & 8730 \\ 0 & 631 \end{array}\right),\left(\begin{array}{rr} 4321 & 7650 \\ 2430 & 4861 \end{array}\right),\left(\begin{array}{rr} 217 & 130 \\ 6265 & 7403 \end{array}\right),\left(\begin{array}{rr} 1081 & 1260 \\ 4500 & 5401 \end{array}\right),\left(\begin{array}{rr} 6241 & 0 \\ 0 & 3121 \end{array}\right),\left(\begin{array}{rr} 1 & 7488 \\ 1800 & 1 \end{array}\right),\left(\begin{array}{rr} 6241 & 3120 \\ 9240 & 3121 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4680 & 1 \end{array}\right),\left(\begin{array}{rr} 9051 & 1460 \\ 8140 & 7491 \end{array}\right),\left(\begin{array}{rr} 577 & 450 \\ 5040 & 7201 \end{array}\right),\left(\begin{array}{rr} 1 & 4680 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9089 & 8910 \\ 8145 & 5309 \end{array}\right),\left(\begin{array}{rr} 37 & 450 \\ 4725 & 5923 \end{array}\right),\left(\begin{array}{rr} 8281 & 1260 \\ 5760 & 5401 \end{array}\right),\left(\begin{array}{rr} 5761 & 0 \\ 0 & 6481 \end{array}\right),\left(\begin{array}{rr} 5201 & 0 \\ 0 & 5201 \end{array}\right),\left(\begin{array}{rr} 7201 & 2970 \\ 0 & 7201 \end{array}\right),\left(\begin{array}{rr} 3601 & 3600 \\ 5760 & 3601 \end{array}\right)$.
The torsion field $K:=\Q(E[9360])$ is a degree-$1391229665280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9360\Z)$.
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 | $4$ | \( 13 \) |
| $13$ | additive | $50$ | \( 256 = 2^{8} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 43264.f
consists of 2 curves linked by isogenies of
degree 5.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-130 +11 \sqrt{130}})\) | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.546617815990272.3 | \(\Z/3\Z\) | not in database |
| $12$ | 12.2.1457471295919600173056.46 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $20$ | 20.4.175873389209659734792706457600000000000000000000.1 | \(\Z/5\Z\) | 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 | ord | add | ord | ss | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 7 | 1 | 1 | 1 | - | 1 | 1,1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | 1 | 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.