Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+125x+3970\)
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(homogenize, simplify) |
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\(y^2z=x^3+125xz^2+3970z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+125x+3970\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1, 64\right) \) | $0.63217039319851751203327361050$ | $\infty$ |
| \( \left(129, 1472\right) \) | $0.77302693357831928196251787760$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1:64:1]\) | $0.63217039319851751203327361050$ | $\infty$ |
| \([129:1472:1]\) | $0.77302693357831928196251787760$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1, 64\right) \) | $0.63217039319851751203327361050$ | $\infty$ |
| \( \left(129, 1472\right) \) | $0.77302693357831928196251787760$ | $\infty$ |
Integral points
\((-9,\pm 46)\), \((-1,\pm 62)\), \((1,\pm 64)\), \((14,\pm 92)\), \((31,\pm 194)\), \((34,\pm 218)\), \((129,\pm 1472)\), \((3969,\pm 250048)\)
\([-9:\pm 46:1]\), \([-1:\pm 62:1]\), \([1:\pm 64:1]\), \([14:\pm 92:1]\), \([31:\pm 194:1]\), \([34:\pm 218:1]\), \([129:\pm 1472:1]\), \([3969:\pm 250048:1]\)
\((-9,\pm 46)\), \((-1,\pm 62)\), \((1,\pm 64)\), \((14,\pm 92)\), \((31,\pm 194)\), \((34,\pm 218)\), \((129,\pm 1472)\), \((3969,\pm 250048)\)
Invariants
| Conductor: | $N$ | = | \( 9200 \) | = | $2^{4} \cdot 5^{2} \cdot 23$ |
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| Minimal Discriminant: | $\Delta$ | = | $-6933708800$ | = | $-1 \cdot 2^{19} \cdot 5^{2} \cdot 23^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{2109375}{67712} \) | = | $2^{-7} \cdot 3^{3} \cdot 5^{7} \cdot 23^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.56829601975126539358703362026$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.39309081288102997826365839007$ |
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| $abc$ quality: | $Q$ | ≈ | $1.2205379736306603$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.297504575878639$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.43942505567111529219380728416$ |
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| Real period: | $\Omega$ | ≈ | $1.0019830575113213992189573691$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $3.5223716866274137711153773461 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.522371687 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.001983 \cdot 0.439425 \cdot 8}{1^2} \\ & \approx 3.522371687\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 12096 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{11}^{*}$ | additive | -1 | 4 | 19 | 7 |
| $5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
| $23$ | $2$ | $I_{2}$ | 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 |
|---|---|---|---|
| $2$ | 2G | 8.2.0.1 | $2$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 8.2.0.a.1, level \( 8 = 2^{3} \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 6 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 7 & 0 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 7 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[8])$ is a degree-$768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 25 = 5^{2} \) |
| $5$ | additive | $10$ | \( 368 = 2^{4} \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 400 = 2^{4} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 9200x consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 1150b1, its twist by $-4$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.200.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | add | ss | add | ord | ord | ord | ord | ord | nonsplit | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 6,2 | - | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 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.