Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-1133x-48363\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-1133xz^2-48363z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-91800x-35532000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(52, 175\right) \) | $2.3496846714574747421780012751$ | $\infty$ |
| \( \left(47, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([52:175:1]\) | $2.3496846714574747421780012751$ | $\infty$ |
| \([47:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(465, 4725\right) \) | $2.3496846714574747421780012751$ | $\infty$ |
| \( \left(420, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(47, 0\right) \), \((52,\pm 175)\), \((1127,\pm 37800)\)
\([47:0:1]\), \([52:\pm 175:1]\), \([1127:\pm 37800:1]\)
\( \left(47, 0\right) \), \((52,\pm 175)\), \((1127,\pm 37800)\)
Invariants
| Conductor: | $N$ | = | \( 9600 \) | = | $2^{7} \cdot 3 \cdot 5^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-933120000000$ | = | $-1 \cdot 2^{14} \cdot 3^{6} \cdot 5^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{628864}{3645} \) | = | $-1 \cdot 2^{7} \cdot 3^{-6} \cdot 5^{-1} \cdot 17^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.98066675709903290180723220365$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.63272390977128681314658493800$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9635631001560863$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8291745696981687$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.3496846714574747421780012751$ |
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| Real period: | $\Omega$ | ≈ | $0.36868277837290389656667123989$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.4651530918926626660019613456 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.465153092 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.368683 \cdot 2.349685 \cdot 16}{2^2} \\ & \approx 3.465153092\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9216 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | 1 | 7 | 14 | 0 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 41 & 4 \\ 82 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 76 & 49 \\ 17 & 114 \end{array}\right),\left(\begin{array}{rr} 117 & 4 \\ 116 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 61 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 74 & 1 \\ 23 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$2949120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 | $2$ | \( 25 = 5^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3200 = 2^{7} \cdot 5^{2} \) |
| $5$ | additive | $18$ | \( 384 = 2^{7} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 9600.l
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1920.d2, its twist by $40$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.46080.4 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.589824000000.7 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.212336640000.22 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.143327232000000.22 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | nonsplit | add | ss | ord | ord | ss | ss | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 1 | - | 1,1 | 1 | 1 | 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 | 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.