Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-12293x+868443\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-12293xz^2+868443z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-995760x+636082200\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2, 945)$ | $0.29831958844719809225594472736$ | $\infty$ |
| $(-137, 0)$ | $0$ | $2$ |
Integral points
\( \left(-137, 0\right) \), \((-101,\pm 1044)\), \((-2,\pm 945)\), \((103,\pm 840)\), \((187,\pm 2268)\), \((1078,\pm 35235)\)
Invariants
| Conductor: | $N$ | = | \( 33600 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7$ |
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| Discriminant: | $\Delta$ | = | $-209991470976000$ | = | $-1 \cdot 2^{10} \cdot 3^{14} \cdot 5^{3} \cdot 7^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{1605176213504}{1640558367} \) | = | $-1 \cdot 2^{14} \cdot 3^{-14} \cdot 7^{-3} \cdot 461^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4436499861875922983510059859$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.46366785761244611351978938471$ |
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| $abc$ quality: | $Q$ | ≈ | $1.212243337747202$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.922507312273219$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.29831958844719809225594472736$ |
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| Real period: | $\Omega$ | ≈ | $0.51177819813718194087816870031$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 168 $ = $ 2\cdot( 2 \cdot 7 )\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.4122853806703741538558283682 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.412285381 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.511778 \cdot 0.298320 \cdot 168}{2^2} \\ & \approx 6.412285381\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 129024 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | -1 | 6 | 10 | 0 |
| $3$ | $14$ | $I_{14}$ | split multiplicative | -1 | 1 | 14 | 14 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 281 & 4 \\ 142 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 62 & 1 \\ 179 & 0 \end{array}\right),\left(\begin{array}{rr} 417 & 4 \\ 416 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 109 & 316 \\ 104 & 315 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 88 & 1 \\ 83 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[420])$ is a degree-$371589120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/420\Z)$.
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 | $2$ | \( 35 = 5 \cdot 7 \) |
| $3$ | split multiplicative | $4$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
| $5$ | additive | $10$ | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
| $7$ | split multiplicative | $8$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 33600.fj
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2100.j2, its twist by $-8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-35}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.2016000.3 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.1084253184000000.51 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.199148544000000.200 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.45349632000000.16 | \(\Z/6\Z\) | 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 | split | add | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | - | 2 | 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 |
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.