Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-500183x-135990888\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-500183xz^2-135990888z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-40514850x-99258901875\)
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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 |
|---|---|---|
| $(817, 175)$ | $5.1546762901940292358625219761$ | $\infty$ |
| $(1113, 26169)$ | $6.9805429930219030546523056755$ | $\infty$ |
| $(-408, 0)$ | $0$ | $2$ |
Integral points
\( \left(-408, 0\right) \), \((817,\pm 175)\), \((1113,\pm 26169)\), \((5992,\pm 460400)\)
Invariants
| Conductor: | $N$ | = | \( 148200 \) | = | $2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $15005250000$ | = | $2^{4} \cdot 3^{5} \cdot 5^{6} \cdot 13 \cdot 19 $ |
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| j-invariant: | $j$ | = | \( \frac{55356847905445888}{60021} \) | = | $2^{11} \cdot 3^{-5} \cdot 13^{-1} \cdot 19^{-1} \cdot 30011^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6715406412387468927452194391$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.63577262483504826897242906533$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0466288639674306$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.281910914712845$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $29.780831620548914140359619646$ |
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| Real period: | $\Omega$ | ≈ | $0.17952182334702313251932967684$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.692618386623245610282852400 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.692618387 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.179522 \cdot 29.780832 \cdot 8}{2^2} \\ & \approx 10.692618387\end{aligned}$$
Modular invariants
Modular form 148200.2.a.k
For more coefficients, see the Downloads section to the right.
| Modular degree: | 737280 |
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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 5 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 | 3 | 4 | 0 |
| $3$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 \( 29640 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 17783 & 0 \\ 0 & 29639 \end{array}\right),\left(\begin{array}{rr} 12601 & 12600 \\ 27430 & 751 \end{array}\right),\left(\begin{array}{rr} 20011 & 20010 \\ 12610 & 15571 \end{array}\right),\left(\begin{array}{rr} 29633 & 8 \\ 29632 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 3656 & 23715 \\ 2285 & 17786 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 29634 & 29635 \end{array}\right),\left(\begin{array}{rr} 3956 & 17785 \\ 13855 & 6 \end{array}\right),\left(\begin{array}{rr} 12176 & 23715 \\ 18725 & 17786 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[29640])$ is a degree-$2379002727628800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/29640\Z)$.
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 | $2$ | \( 18525 = 3 \cdot 5^{2} \cdot 13 \cdot 19 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 49400 = 2^{3} \cdot 5^{2} \cdot 13 \cdot 19 \) |
| $5$ | additive | $14$ | \( 1976 = 2^{3} \cdot 13 \cdot 19 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 11400 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 148200.k
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 5928.m3, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{741}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1235}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{741})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | 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 | ss | ord | nonsplit | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | - | 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.