Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-32160x-791820\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-32160xz^2-791820z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2604987x-569421846\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-117, 1176)$ | $2.6560254207609331978095398023$ | $\infty$ |
| $(-166, 0)$ | $0$ | $2$ |
Integral points
\( \left(-166, 0\right) \), \((-117,\pm 1176)\), \((858,\pm 24576)\)
Invariants
| Conductor: | $N$ | = | \( 11760 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $1865262437498880$ | = | $2^{24} \cdot 3^{3} \cdot 5 \cdot 7^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{7633736209}{3870720} \) | = | $2^{-12} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{3} \cdot 179^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6226749432295265656809963540$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.043427311858075396288912139180$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9780810097967987$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.561136974500809$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.6560254207609331978095398023$ |
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| Real period: | $\Omega$ | ≈ | $0.37615494170437487485925664532$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2^{2}\cdot3\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.9944625238699994529908534310 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.994462524 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.376155 \cdot 2.656025 \cdot 24}{2^2} \\ & \approx 5.994462524\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 55296 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{16}^{*}$ | additive | -1 | 4 | 24 | 12 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 |
|---|---|---|
| $2$ | 2B | 4.12.0.8 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 16 & 423 \\ 521 & 194 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 374 & 11 \end{array}\right),\left(\begin{array}{rr} 623 & 816 \\ 138 & 833 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 352 & 3 \\ 717 & 754 \end{array}\right),\left(\begin{array}{rr} 344 & 819 \\ 645 & 374 \end{array}\right),\left(\begin{array}{rr} 817 & 24 \\ 816 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 249 & 388 \\ 500 & 629 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$185794560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $2$ | \( 735 = 3 \cdot 5 \cdot 7^{2} \) |
| $3$ | split multiplicative | $4$ | \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \) |
| $5$ | split multiplicative | $6$ | \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 11760co
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 210a1, its twist by $28$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-105}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{105})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.4630500.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{7})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{-15})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.18151560000.2 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.343064484000000.54 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1734805094400.6 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.1792336896000000.27 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.31116960000.7 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.4.343064484000000.14 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.6.130735066580434673139791580364800000000.2 | \(\Z/18\Z\) | 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 | split | split | add | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | 4 | - | 1,1 | 1 | 3 | 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 | 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.