Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-3798x-91615\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-3798xz^2-91615z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4922235x-4200547626\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{141}{4}, \frac{137}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-282:137:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-1254, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 5054 \) | = | $2 \cdot 7 \cdot 19^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $4610496338$ | = | $2 \cdot 7^{2} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{128787625}{98} \) | = | $2^{-1} \cdot 5^{3} \cdot 7^{-2} \cdot 101^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.78670818133819499619005083386$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.68551130824502523381446288208$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9676277689392396$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.261325624055992$ | |||
| 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: | $\Omega$ | ≈ | $0.60817709098151674228887984272$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $5.4735938188336506805999185845 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 5.473593819 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.608177 \cdot 1.000000 \cdot 4}{2^2} \\ & \approx 5.473593819\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4752 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $19$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.6.0.6 | $6$ |
| $3$ | 3B | 9.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7183 & 2052 \\ 4294 & 1369 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 1825 & 2052 \\ 1938 & 2281 \end{array}\right),\left(\begin{array}{rr} 9541 & 36 \\ 9540 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 6139 \end{array}\right),\left(\begin{array}{rr} 1540 & 1539 \\ 57 & 7354 \end{array}\right),\left(\begin{array}{rr} 8567 & 0 \\ 0 & 9575 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right),\left(\begin{array}{rr} 5321 & 2052 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9576])$ is a degree-$1715626967040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9576\Z)$.
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 | $4$ | \( 361 = 19^{2} \) |
| $7$ | split multiplicative | $8$ | \( 722 = 2 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 14 = 2 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 5054c
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14a6, its twist by $-19$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-19}) \) | \(\Z/6\Z\) | 2.0.19.1-196.2-a5 |
| $4$ | \(\Q(\sqrt{-38 +38 \sqrt{-7}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-19})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.7114374288.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.16468459.1 | \(\Z/18\Z\) | not in database |
| $8$ | 8.0.20506261651456.52 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.26783688687616.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.320410338304.2 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.21080517080281344.1 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.71096111421110616064.1 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 7 | 19 |
|---|---|---|---|---|
| Reduction type | split | ord | split | add |
| $\lambda$-invariant(s) | 2 | 2 | 1 | - |
| $\mu$-invariant(s) | 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 $0$.