Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+1567x+88737\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+1567xz^2+88737z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+126900x+65070000\)
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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 |
|---|---|---|
| $(-23, 200)$ | $1.0746837430901291003967206688$ | $\infty$ |
| $(57, 600)$ | $1.4371115438740419989520498105$ | $\infty$ |
| $(-33, 0)$ | $0$ | $2$ |
Integral points
\( \left(-33, 0\right) \), \((-23,\pm 200)\), \((-8,\pm 275)\), \((16,\pm 343)\), \((57,\pm 600)\), \((67,\pm 700)\), \((111,\pm 1272)\), \((163,\pm 2156)\), \((457,\pm 9800)\), \((16777,\pm 2173000)\)
Invariants
| Conductor: | $N$ | = | \( 33600 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7$ |
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| Discriminant: | $\Delta$ | = | $-3687936000000$ | = | $-1 \cdot 2^{15} \cdot 3 \cdot 5^{6} \cdot 7^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{830584}{7203} \) | = | $2^{3} \cdot 3^{-1} \cdot 7^{-4} \cdot 47^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0926593185103111219543467911$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.57849361340667070211757302733$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9710857330134317$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.4850092039247147$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.5401288166463071596497014440$ |
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| Real period: | $\Omega$ | ≈ | $0.57647681283617296394353133349$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot1\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!$ | ≈ | $7.1027884126192780835009688160 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.102788413 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.576477 \cdot 1.540129 \cdot 32}{2^2} \\ & \approx 7.102788413\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 65536 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 6 | 15 | 0 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 736 & 675 \\ 565 & 506 \end{array}\right),\left(\begin{array}{rr} 833 & 8 \\ 832 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 104 & 685 \\ 815 & 114 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 503 & 0 \\ 0 & 839 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 241 & 680 \\ 460 & 201 \end{array}\right),\left(\begin{array}{rr} 281 & 780 \\ 110 & 191 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 834 & 835 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$1486356480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $4$ | \( 75 = 3 \cdot 5^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 11200 = 2^{6} \cdot 5^{2} \cdot 7 \) |
| $5$ | additive | $14$ | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 33600er
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 672e4, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{10})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7644119040000.7 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.56646696960000.4 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.4588382453760000.17 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | nonsplit | add | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 4 | - | 2 | 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 |
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.