Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-271692x+2804976\)
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(homogenize, simplify) |
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\(y^2z=x^3-271692xz^2+2804976z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-271692x+2804976\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(957, 24885)$ | $4.8982721184563944342723078645$ | $\infty$ |
| $(516, 0)$ | $0$ | $2$ |
Integral points
\( \left(516, 0\right) \), \((957,\pm 24885)\)
Invariants
| Conductor: | $N$ | = | \( 31680 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $1280144377847808000$ | = | $2^{16} \cdot 3^{6} \cdot 5^{3} \cdot 11^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{46424454082884}{26794860125} \) | = | $2^{2} \cdot 3^{3} \cdot 5^{-3} \cdot 11^{-8} \cdot 7547^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1639856987788853716959579467$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.69048331369823678010869249963$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1395461376610276$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.742717869061533$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.8982721184563944342723078645$ |
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| Real period: | $\Omega$ | ≈ | $0.23129871477595771642029207916$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2\cdot2\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.7977842737312304709993154719 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.797784274 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.231299 \cdot 4.898272 \cdot 24}{2^2} \\ & \approx 6.797784274\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 589824 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}^{*}$ | additive | 1 | 6 | 16 | 0 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
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 | 8.12.0.10 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 1759 & 0 \\ 0 & 2639 \end{array}\right),\left(\begin{array}{rr} 1201 & 1776 \\ 2568 & 1009 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2625 & 16 \\ 2624 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 166 \\ 2354 & 2115 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 8 & 65 \end{array}\right),\left(\begin{array}{rr} 718 & 3 \\ 621 & 1780 \end{array}\right),\left(\begin{array}{rr} 661 & 1986 \\ 120 & 2371 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1313 & 2190 \\ 1398 & 1337 \end{array}\right)$.
The torsion field $K:=\Q(E[2640])$ is a degree-$38928384000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2640\Z)$.
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 | $2$ | \( 45 = 3^{2} \cdot 5 \) |
| $3$ | additive | $6$ | \( 704 = 2^{6} \cdot 11 \) |
| $5$ | split multiplicative | $6$ | \( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 31680.dw
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 440.c1, its twist by $-24$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{30}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.82944000000.31 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.5184000000.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.530841600.4 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.2098454003712.15 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | 16.0.176120502681600000000.3 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | add | add | split | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord |
| $\lambda$-invariant(s) | - | - | 2 | 1 | 1 | 1 | 1 | 3 | 1 | 3 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.