Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-633x+11488\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-633xz^2+11488z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-51300x+8528625\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-7, 125\right) \) | $0.75697733532268647102074573111$ | $\infty$ |
| \( \left(13, 75\right) \) | $1.3929944852780169161860922216$ | $\infty$ |
| \( \left(-32, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-7:125:1]\) | $0.75697733532268647102074573111$ | $\infty$ |
| \([13:75:1]\) | $1.3929944852780169161860922216$ | $\infty$ |
| \([-32:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-60, 3375\right) \) | $0.75697733532268647102074573111$ | $\infty$ |
| \( \left(120, 2025\right) \) | $1.3929944852780169161860922216$ | $\infty$ |
| \( \left(-285, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-32, 0\right) \), \((-13,\pm 133)\), \((-7,\pm 125)\), \((13,\pm 75)\), \((17,\pm 77)\), \((44,\pm 266)\), \((63,\pm 475)\), \((93,\pm 875)\), \((1413,\pm 53125)\), \((1868,\pm 80750)\)
\([-32:0:1]\), \([-13:\pm 133:1]\), \([-7:\pm 125:1]\), \([13:\pm 75:1]\), \([17:\pm 77:1]\), \([44:\pm 266:1]\), \([63:\pm 475:1]\), \([93:\pm 875:1]\), \([1413:\pm 53125:1]\), \([1868:\pm 80750:1]\)
\( \left(-32, 0\right) \), \((-13,\pm 133)\), \((-7,\pm 125)\), \((13,\pm 75)\), \((17,\pm 77)\), \((44,\pm 266)\), \((63,\pm 475)\), \((93,\pm 875)\), \((1413,\pm 53125)\), \((1868,\pm 80750)\)
Invariants
| Conductor: | $N$ | = | \( 36100 \) | = | $2^{2} \cdot 5^{2} \cdot 19^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-42868750000$ | = | $-1 \cdot 2^{4} \cdot 5^{8} \cdot 19^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{16384}{25} \) | = | $-1 \cdot 2^{14} \cdot 5^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.73167213120444117093635261110$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0402056299908675678386946206$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8244854285178197$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.073901992000791$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0175260653448193580946185157$ |
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| Real period: | $\Omega$ | ≈ | $1.0253955813421985430083361211$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 3\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $6.2602003878305456884048836198 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.260200388 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.025396 \cdot 1.017526 \cdot 24}{2^2} \\ & \approx 6.260200388\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23040 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $19$ | $2$ | $III$ | additive | 1 | 2 | 3 | 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 | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 380 = 2^{2} \cdot 5 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 77 & 4 \\ 154 & 9 \end{array}\right),\left(\begin{array}{rr} 84 & 1 \\ 359 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 377 & 4 \\ 376 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 289 & 96 \\ 284 & 95 \end{array}\right)$.
The torsion field $K:=\Q(E[380])$ is a degree-$472780800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/380\Z)$.
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 | $2$ | \( 475 = 5^{2} \cdot 19 \) |
| $5$ | additive | $18$ | \( 1444 = 2^{2} \cdot 19^{2} \) |
| $19$ | additive | $110$ | \( 100 = 2^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 36100.b
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 7220.g2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-19}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{19 +2 \sqrt{95}})\) | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.120437455360000.37 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.120437455360000.19 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | ord | add | ss | ord | ord | ord | add | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | - | 2,6 | 2 | 2 | 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 | 0,0 |
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.