Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-142x-609\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-142xz^2-609z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2275x-41250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(94, 853\right) \) | $3.0019061106373577523740732842$ | $\infty$ |
| \( \left(-6, 3\right) \) | $0$ | $2$ |
| \( \left(14, -7\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([94:853:1]\) | $3.0019061106373577523740732842$ | $\infty$ |
| \([-6:3:1]\) | $0$ | $2$ |
| \([14:-7:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(375, 7200\right) \) | $3.0019061106373577523740732842$ | $\infty$ |
| \( \left(-25, 0\right) \) | $0$ | $2$ |
| \( \left(55, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-6, 3\right) \), \( \left(14, -7\right) \), \( \left(94, 853\right) \), \( \left(94, -947\right) \)
\([-6:3:1]\), \([14:-7:1]\), \([94:853:1]\), \([94:-947:1]\)
\( \left(-25, 0\right) \), \( \left(55, 0\right) \), \((375,\pm 7200)\)
Invariants
| Conductor: | $N$ | = | \( 425 \) | = | $5^{2} \cdot 17$ |
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| Minimal Discriminant: | $\Delta$ | = | $4515625$ | = | $5^{6} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{20346417}{289} \) | = | $3^{3} \cdot 7^{3} \cdot 13^{3} \cdot 17^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.081509244995734051634330601653$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.72320971122131613566604906496$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0296285559493121$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.37618186396722$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0019061106373577523740732842$ |
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| Real period: | $\Omega$ | ≈ | $1.3837501982215705488562905972$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.0769440878184938402967154199 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.076944088 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.383750 \cdot 3.001906 \cdot 8}{4^2} \\ & \approx 2.076944088\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 64 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $17$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 |
|---|---|---|---|
| $2$ | 2Cs | 16.48.0.20 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2720 = 2^{5} \cdot 5 \cdot 17 \), index $1536$, genus $53$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 141 & 2200 \\ 810 & 1461 \end{array}\right),\left(\begin{array}{rr} 543 & 0 \\ 0 & 2719 \end{array}\right),\left(\begin{array}{rr} 1981 & 2260 \\ 1880 & 1801 \end{array}\right),\left(\begin{array}{rr} 511 & 2310 \\ 1360 & 1 \end{array}\right),\left(\begin{array}{rr} 2689 & 32 \\ 2688 & 33 \end{array}\right),\left(\begin{array}{rr} 9 & 32 \\ 2664 & 2521 \end{array}\right),\left(\begin{array}{rr} 29 & 8 \\ 1532 & 1173 \end{array}\right)$.
The torsion field $K:=\Q(E[2720])$ is a degree-$9625927680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2720\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 25 = 5^{2} \) |
| $5$ | additive | $14$ | \( 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 25 = 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 425a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 17a2, its twist by $5$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{5}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | 4.4.7225.1-289.1-c3 |
| $4$ | \(\Q(i, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.13363360000.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.114162766875.4 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 |
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 | ord | ss | add | ord | ss | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 2 | 1,1 | - | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 1 | 0,0 | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.