Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-7370942x+8846283341\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-7370942xz^2+8846283341z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-117935067x+566044198774\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2721, 93139)$ | $0$ | $6$ |
Integral points
\( \left(721, 62139\right) \), \( \left(721, -62861\right) \), \( \left(2721, 93139\right) \), \( \left(2721, -95861\right) \)
Invariants
| Conductor: | $N$ | = | \( 8190 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-8162806640625000000000$ | = | $-1 \cdot 2^{9} \cdot 3^{8} \cdot 5^{18} \cdot 7^{2} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{60752633741424905775769}{11197265625000000000} \) | = | $-1 \cdot 2^{-9} \cdot 3^{-2} \cdot 5^{-18} \cdot 7^{-2} \cdot 13^{-1} \cdot 31^{3} \cdot 1268119^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9291416315687398318485071271$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.3798354872346849861508845086$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0187317206577513$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.584333994521741$ | |||
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: | $\Omega$ | ≈ | $0.12593458829181279599755303393$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1296 $ = $ 3^{2}\cdot2^{2}\cdot( 2 \cdot 3^{2} )\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.5336451785052606559119092213 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 4.533645179 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.125935 \cdot 1.000000 \cdot 1296}{6^2} \\ & \approx 4.533645179\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 746496 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $18$ | $I_{18}$ | split multiplicative | -1 | 1 | 18 | 18 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $1$ | $I_{1}$ | split 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B.1.1 | 9.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 32760 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19657 & 36 \\ 10 & 361 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 32725 & 36 \\ 32724 & 37 \end{array}\right),\left(\begin{array}{rr} 16390 & 27 \\ 7447 & 30754 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 16689 & 298 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 6139 \end{array}\right),\left(\begin{array}{rr} 3635 & 32724 \\ 7486 & 1483 \end{array}\right),\left(\begin{array}{rr} 27724 & 9 \\ 15199 & 178 \end{array}\right),\left(\begin{array}{rr} 9385 & 36 \\ 11514 & 9841 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[32760])$ is a degree-$175294937825280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/32760\Z)$.
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 | $4$ | \( 117 = 3^{2} \cdot 13 \) |
| $3$ | additive | $8$ | \( 91 = 7 \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 8190bz
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 2730p6, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-26}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $3$ | 3.3.670761.2 | \(\Z/18\Z\) | not in database |
| $4$ | 4.2.4586400.5 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.12147848616267.3 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.1466954307.20 | \(\Z/18\Z\) | not in database |
| $6$ | 6.0.2994669644069376.2 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/36\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.1306849698240310085247366316592533934392827.5 | \(\Z/3\Z \oplus \Z/18\Z\) | 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 | 13 |
|---|---|---|---|---|---|
| Reduction type | split | add | split | split | split |
| $\lambda$-invariant(s) | 2 | - | 1 | 3 | 1 |
| $\mu$-invariant(s) | 1 | - | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
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 $0$.