Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-28782451x-43764761202\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-28782451xz^2-43764761202z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-37302055875x-2041776792461250\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 48050 \) | = | $2 \cdot 5^{2} \cdot 31^{2}$ |
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| Discriminant: | $\Delta$ | = | $698688337871667200000000$ | = | $2^{21} \cdot 5^{8} \cdot 31^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{7915045705}{2097152} \) | = | $2^{-21} \cdot 5 \cdot 7^{3} \cdot 31 \cdot 53^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2838358185620075320751231865$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.078447592717490214944825918677$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9559894939860223$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.857091594081697$ | |||
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: | $\Omega$ | ≈ | $0.066466620825079629347075315582$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3 $ = $ 1\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.19939986247523888804122594674 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.199399862 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.066467 \cdot 1.000000 \cdot 3}{1^2} \\ & \approx 0.199399862\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9374400 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{21}$ | nonsplit multiplicative | 1 | 1 | 21 | 21 |
| $5$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $31$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 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 |
|---|---|---|
| $2$ | 2G | 8.2.0.2 |
| $3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.16.0-24.b.1.4, level \( 24 = 2^{3} \cdot 3 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 2 & 3 \\ 7 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 22 \\ 14 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 6 \\ 15 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19 & 6 \\ 18 & 7 \end{array}\right)$.
The torsion field $K:=\Q(E[24])$ is a degree-$4608$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 24025 = 5^{2} \cdot 31^{2} \) |
| $3$ | good | $2$ | \( 24025 = 5^{2} \cdot 31^{2} \) |
| $5$ | additive | $14$ | \( 1922 = 2 \cdot 31^{2} \) |
| $7$ | good | $2$ | \( 24025 = 5^{2} \cdot 31^{2} \) |
| $31$ | additive | $362$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 48050.b
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 48050.p1, its twist by $-155$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.3.192200.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.648675.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.295526720000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.1262337766875.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.997402680000.3 | \(\Z/6\Z\) | not in database |
| $9$ | 9.3.139749879624984000000.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.6006399343075824213089699938107000000000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.527310779090332989900879006912000000000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.9999374773861129289972224131072000000000000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 31 |
|---|---|---|---|---|
| Reduction type | nonsplit | ord | add | add |
| $\lambda$-invariant(s) | 2 | 2 | - | - |
| $\mu$-invariant(s) | 0 | 1 | - | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.