Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-12250x+465625\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-12250xz^2+465625z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15876675x+21962346750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(80, 35)$ | $2.8119860777888197263212882786$ | $\infty$ |
| $(-120, 535)$ | $3.6112137343996122942922805911$ | $\infty$ |
Integral points
\( \left(-120, 535\right) \), \( \left(-120, -415\right) \), \( \left(80, 35\right) \), \( \left(80, -115\right) \)
Invariants
| Conductor: | $N$ | = | \( 138675 \) | = | $3 \cdot 5^{2} \cdot 43^{2}$ |
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| Discriminant: | $\Delta$ | = | $21938818359375$ | = | $3^{5} \cdot 5^{11} \cdot 43^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{7037694889}{759375} \) | = | $3^{-5} \cdot 5^{-5} \cdot 43 \cdot 547^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2939607479128358309164452092$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.13762489425314142696274154297$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9179627623362381$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.366043079792992$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $10.041526879495029821854764252$ |
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| Real period: | $\Omega$ | ≈ | $0.65804086623810079178496531033$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $13.215470092272165134813826677 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.215470092 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.658041 \cdot 10.041527 \cdot 2}{1^2} \\ & \approx 13.215470092\end{aligned}$$
Modular invariants
Modular form 138675.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 403200 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $5$ | $2$ | $I_{5}^{*}$ | additive | 1 | 2 | 11 | 5 |
| $43$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
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 label 60.2.0.a.1, level \( 60 = 2^{2} \cdot 3 \cdot 5 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 37 & 2 \\ 37 & 3 \end{array}\right),\left(\begin{array}{rr} 59 & 2 \\ 58 & 3 \end{array}\right),\left(\begin{array}{rr} 31 & 2 \\ 31 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 59 & 0 \end{array}\right),\left(\begin{array}{rr} 41 & 2 \\ 41 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[60])$ is a degree-$1105920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/60\Z)$.
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 |
|---|---|---|---|
| $3$ | nonsplit multiplicative | $4$ | \( 46225 = 5^{2} \cdot 43^{2} \) |
| $5$ | additive | $14$ | \( 1849 = 43^{2} \) |
| $43$ | additive | $338$ | \( 75 = 3 \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 138675u consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 27735m1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.110940.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.738461016000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | 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 | ord | nonsplit | add | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | add | ord |
| $\lambda$-invariant(s) | 3 | 6 | - | 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.