Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-11566x+597700\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-11566xz^2+597700z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-14989563x+27931259862\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(6, 724)$ | $4.2116492756295074463455856128$ | $\infty$ |
| $(24, 566)$ | $0$ | $3$ |
Integral points
\( \left(6, 724\right) \), \( \left(6, -730\right) \), \( \left(24, 566\right) \), \( \left(24, -590\right) \), \( \left(5226, 375110\right) \), \( \left(5226, -380336\right) \)
Invariants
| Conductor: | $N$ | = | \( 2890 \) | = | $2 \cdot 5 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-55806059528000$ | = | $-1 \cdot 2^{6} \cdot 5^{3} \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{24529249}{8000} \) | = | $-1 \cdot 2^{-6} \cdot 5^{-3} \cdot 17 \cdot 113^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3502567874885364484285685943$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.53855210854894093840445448428$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8867299725112243$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.035512263824783$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.2116492756295074463455856128$ |
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| Real period: | $\Omega$ | ≈ | $0.59331219122446151073820104063$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 18 $ = $ ( 2 \cdot 3 )\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.9976457207853182526987273263 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.997645721 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.593312 \cdot 4.211649 \cdot 18}{3^2} \\ & \approx 4.997645721\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7344 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $17$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 60.16.0-60.a.1.8, level \( 60 = 2^{2} \cdot 3 \cdot 5 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 37 & 6 \\ 51 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 6 & 1 \\ 25 & 51 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 31 & 6 \\ 33 & 19 \end{array}\right),\left(\begin{array}{rr} 55 & 6 \\ 54 & 7 \end{array}\right),\left(\begin{array}{rr} 57 & 58 \\ 50 & 53 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[60])$ is a degree-$138240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/60\Z)$.
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 | $4$ | \( 1445 = 5 \cdot 17^{2} \) |
| $3$ | good | $2$ | \( 289 = 17^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 578 = 2 \cdot 17^{2} \) |
| $17$ | additive | $114$ | \( 10 = 2 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 2890.p
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 2890.o1, its twist by $17$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.5780.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.668168000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.2255067.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.30406385320128.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.733936223705176688832000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | ord | nonsplit | ord | ss | ord | add | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 5 | 1 | 3 | 1,1 | 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.