Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-22125x+1268125\)
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(homogenize, simplify) |
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\(y^2z=x^3-22125xz^2+1268125z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-22125x+1268125\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(89, 63\right) \) | $0.78514883347465900877021646497$ | $\infty$ |
| \( \left(75, 175\right) \) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([89:63:1]\) | $0.78514883347465900877021646497$ | $\infty$ |
| \([75:175:1]\) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(89, 63\right) \) | $0.78514883347465900877021646497$ | $\infty$ |
| \( \left(75, 175\right) \) | $0$ | $3$ |
Integral points
\((-100,\pm 1575)\), \((-9,\pm 1211)\), \((75,\pm 175)\), \((89,\pm 63)\), \((125,\pm 675)\)
\([-100:\pm 1575:1]\), \([-9:\pm 1211:1]\), \([75:\pm 175:1]\), \([89:\pm 63:1]\), \([125:\pm 675:1]\)
\((-100,\pm 1575)\), \((-9,\pm 1211)\), \((75,\pm 175)\), \((89,\pm 63)\), \((125,\pm 675)\)
Invariants
| Conductor: | $N$ | = | \( 6300 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1562793750000$ | = | $-1 \cdot 2^{4} \cdot 3^{6} \cdot 5^{8} \cdot 7^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{262885120}{343} \) | = | $-1 \cdot 2^{8} \cdot 5 \cdot 7^{-3} \cdot 59^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2468452953784009657418824069$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.60646851743170256616199047420$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8938173325222778$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.75854990580215$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.78514883347465900877021646497$ |
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| Real period: | $\Omega$ | ≈ | $0.84414910454180434586373797844$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 54 $ = $ 3\cdot2\cdot3\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9766961082580539545119256120 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.976696108 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.844149 \cdot 0.785149 \cdot 54}{3^2} \\ & \approx 3.976696108\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 12960 |
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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 4 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 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3B.1.1 | 3.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 42.16.0-42.a.1.4, level \( 42 = 2 \cdot 3 \cdot 7 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 4 & 33 \\ 11 & 35 \end{array}\right),\left(\begin{array}{rr} 37 & 6 \\ 36 & 7 \end{array}\right),\left(\begin{array}{rr} 39 & 40 \\ 32 & 35 \end{array}\right),\left(\begin{array}{rr} 31 & 6 \\ 9 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[42])$ is a degree-$36288$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/42\Z)$.
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 | $2$ | \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \) |
| $3$ | additive | $6$ | \( 100 = 2^{2} \cdot 5^{2} \) |
| $5$ | additive | $14$ | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
| $7$ | split multiplicative | $8$ | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 6300bd
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 700b2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.175.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.214375.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.21870000.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.187570506627000000.6 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.1230650093979747000000000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | add | split | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | - | 2 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 1 |
| $\mu$-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.