Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+304875x+31843125\)
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(homogenize, simplify) |
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\(y^2z=x^3+304875xz^2+31843125z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+304875x+31843125\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(900, 32175)$ | $0.24673806268144936411050547125$ | $\infty$ |
| $(471, 16731)$ | $0.56227178026113663795839496958$ | $\infty$ |
Integral points
\((-101,\pm 143)\), \((-75,\pm 2925)\), \((-36,\pm 4563)\), \((75,\pm 7425)\), \((141,\pm 8811)\), \((225,\pm 10575)\), \((471,\pm 16731)\), \((900,\pm 32175)\), \((1329,\pm 52767)\), \((1875,\pm 84825)\), \((3175,\pm 181675)\), \((4761,\pm 330759)\), \((11625,\pm 1254825)\), \((14100,\pm 1675575)\), \((36975,\pm 7110675)\), \((60300,\pm 14807925)\), \((1376131,\pm 1614320279)\)
Invariants
| Conductor: | $N$ | = | \( 128700 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-2251657666293750000$ | = | $-1 \cdot 2^{4} \cdot 3^{6} \cdot 5^{8} \cdot 11^{3} \cdot 13^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{687830780160}{494190983} \) | = | $2^{8} \cdot 3^{3} \cdot 5 \cdot 11^{-3} \cdot 13^{-5} \cdot 271^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2099087656342680838184379325$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.35659495282416455191456505140$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9808008879990033$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.207018045571571$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.12948631526326158028429796755$ |
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| Real period: | $\Omega$ | ≈ | $0.16490209794924827931949424279$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 540 $ = $ 3\cdot2^{2}\cdot3\cdot3\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $11.530385123019985948043093552 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.530385123 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.164902 \cdot 0.129486 \cdot 540}{1^2} \\ & \approx 11.530385123\end{aligned}$$
Modular invariants
Modular form 128700.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1900800 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $11$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 286 = 2 \cdot 11 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 67 & 2 \\ 67 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 285 & 0 \end{array}\right),\left(\begin{array}{rr} 79 & 2 \\ 79 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 285 & 2 \\ 284 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[286])$ is a degree-$1037836800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/286\Z)$.
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 | $2$ | \( 32175 = 3^{2} \cdot 5^{2} \cdot 11 \cdot 13 \) |
| $3$ | additive | $6$ | \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \) |
| $5$ | additive | $14$ | \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 11700 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 128700ci consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 14300a1, its twist by $-15$.
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 |
|---|---|---|---|
| $3$ | 3.1.3575.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1827629375.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.624629070000.2 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | add | add | ord | split | split | ord | ord | ord | ord | ord | ord | ord | ss | ord |
| $\lambda$-invariant(s) | - | - | - | 2 | 3 | 3 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2,2 | 2 |
| $\mu$-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.