Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-6001x+178052\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-6001xz^2+178052z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7776675x+8330535774\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-5, 458\right) \) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-5:458:1]\) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-177, 98496\right) \) | $0$ | $6$ |
Integral points
\( \left(-5, 458\right) \), \( \left(-5, -454\right) \), \( \left(43, -22\right) \), \( \left(52, 59\right) \), \( \left(52, -112\right) \)
\([-5:458:1]\), \([-5:-454:1]\), \([43:-22:1]\), \([52:59:1]\), \([52:-112:1]\)
\((-177,\pm 98496)\), \( \left(1551, 0\right) \), \((1875,\pm 18468)\)
Invariants
| Conductor: | $N$ | = | \( 4902 \) | = | $2 \cdot 3 \cdot 19 \cdot 43$ |
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| Minimal Discriminant: | $\Delta$ | = | $55042322688$ | = | $2^{8} \cdot 3^{6} \cdot 19^{3} \cdot 43 $ |
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| j-invariant: | $j$ | = | \( \frac{23894093340015625}{55042322688} \) | = | $2^{-8} \cdot 3^{-6} \cdot 5^{6} \cdot 19^{-3} \cdot 41^{3} \cdot 43^{-1} \cdot 281^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.94194491066244646766645142455$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.94194491066244646766645142455$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9843247587061957$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.438112134942098$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $1.1206877006861502970918967228$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 36 $ = $ 2\cdot( 2 \cdot 3 )\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.1206877006861502970918967228 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.120687701 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.120688 \cdot 1.000000 \cdot 36}{6^2} \\ & \approx 1.120687701\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6912 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $19$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $43$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $3$ | 3B.1.1 | 3.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9804 = 2^{2} \cdot 3 \cdot 19 \cdot 43 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 9754 & 9795 \end{array}\right),\left(\begin{array}{rr} 6545 & 2 \\ 6540 & 1 \end{array}\right),\left(\begin{array}{rr} 9793 & 12 \\ 9792 & 13 \end{array}\right),\left(\begin{array}{rr} 6394 & 3 \\ 4305 & 9796 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 4095 & 820 \\ 4058 & 809 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8266 & 3 \\ 5133 & 9796 \end{array}\right)$.
The torsion field $K:=\Q(E[9804])$ is a degree-$19723753082880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9804\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 817 = 19 \cdot 43 \) |
| $3$ | split multiplicative | $4$ | \( 86 = 2 \cdot 43 \) |
| $19$ | split multiplicative | $20$ | \( 258 = 2 \cdot 3 \cdot 43 \) |
| $43$ | split multiplicative | $44$ | \( 114 = 2 \cdot 3 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 4902h
consists of 4 curves linked by isogenies of
degrees dividing 6.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{817}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | 4.0.7353.2 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.1476922032.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.36088866774801.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $9$ | 9.3.36926167171241565786672.1 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.6.1374901366890672405667171134473070725220993584303394148608.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 19 | 43 |
|---|---|---|---|---|
| Reduction type | nonsplit | split | split | split |
| $\lambda$-invariant(s) | 1 | 3 | 1 | 3 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.