Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+5164867x-6108845637\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+5164867xz^2-6108845637z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+418354200x-4454603532000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(993, 0)$ | $0$ | $2$ |
Integral points
\( \left(993, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 124800 \) | = | $2^{7} \cdot 3 \cdot 5^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-24948202500000000000000$ | = | $-1 \cdot 2^{14} \cdot 3^{10} \cdot 5^{16} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{59519748892520576}{97453916015625} \) | = | $2^{7} \cdot 3^{-10} \cdot 5^{-10} \cdot 11^{3} \cdot 13^{-2} \cdot 7043^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9826863884674249968250807221$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3692957215971052818712635805$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0215004882943675$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.992959012291013$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.062920456371253629551286367561$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 160 $ = $ 2\cdot( 2 \cdot 5 )\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.5168182548501451820514547024 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.516818255 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.062920 \cdot 1.000000 \cdot 160}{2^2} \\ & \approx 2.516818255\end{aligned}$$
Modular invariants
Modular form 124800.2.a.dc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8601600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | -1 | 7 | 14 | 0 |
| $3$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $5$ | $4$ | $I_{10}^{*}$ | additive | 1 | 2 | 16 | 10 |
| $13$ | $2$ | $I_{2}$ | split 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 2641 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 2081 & 8 \\ 1042 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 779 & 3118 \\ 0 & 3119 \end{array}\right),\left(\begin{array}{rr} 623 & 3114 \\ 0 & 3119 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 391 & 8 \\ 2145 & 1 \end{array}\right),\left(\begin{array}{rr} 3113 & 8 \\ 3112 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[3120])$ is a degree-$309162147840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3120\Z)$.
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 | $2$ | \( 25 = 5^{2} \) |
| $3$ | split multiplicative | $4$ | \( 41600 = 2^{7} \cdot 5^{2} \cdot 13 \) |
| $5$ | additive | $14$ | \( 1664 = 2^{7} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 9600 = 2^{7} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 124800.dc
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 24960.b2, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.4326400.3 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.35884892160000.30 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.74870947840000.6 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | \(\Z/8\Z\) | 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 | 13 |
|---|---|---|---|---|
| Reduction type | add | split | add | split |
| $\lambda$-invariant(s) | - | 1 | - | 1 |
| $\mu$-invariant(s) | - | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.