Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-833x+9213\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-833xz^2+9213z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-67500x+6918750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(12, 33)$ | $2.8596508697466866586107841572$ | $\infty$ |
Integral points
\((12,\pm 33)\)
Invariants
| Conductor: | $N$ | = | \( 4800 \) | = | $2^{6} \cdot 3 \cdot 5^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1875000000$ | = | $-1 \cdot 2^{6} \cdot 3 \cdot 5^{10} $ |
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| j-invariant: | $j$ | = | \( -\frac{102400}{3} \) | = | $-1 \cdot 2^{12} \cdot 3^{-1} \cdot 5^{2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.55819421240235002732380379608$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1295776382393729395521117090$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0439051029424253$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.756240575699233$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.8596508697466866586107841572$ |
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| Real period: | $\Omega$ | ≈ | $1.4766359094697852559566529894$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.2226631628144610696665204716 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.222663163 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.476636 \cdot 2.859651 \cdot 1}{1^2} \\ & \approx 4.222663163\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2400 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | -1 | 6 | 6 | 0 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 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 |
|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 117 & 110 \\ 100 & 101 \end{array}\right),\left(\begin{array}{rr} 111 & 10 \\ 110 & 11 \end{array}\right),\left(\begin{array}{rr} 6 & 5 \\ 5 & 26 \end{array}\right),\left(\begin{array}{rr} 99 & 110 \\ 100 & 109 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 65 & 1 \end{array}\right),\left(\begin{array}{rr} 89 & 0 \\ 0 & 119 \end{array}\right),\left(\begin{array}{rr} 114 & 107 \\ 115 & 119 \end{array}\right),\left(\begin{array}{rr} 59 & 0 \\ 0 & 119 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$737280$ 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 | $2$ | \( 75 = 3 \cdot 5^{2} \) |
| $3$ | split multiplicative | $4$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
| $5$ | additive | $2$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 4800.br
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 75.c1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.300.1 | \(\Z/2\Z\) | not in database |
| $4$ | 4.4.8000.1 | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.270000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.11337408000000.9 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.2099520000000000.32 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | 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 | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 |
| $\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.