Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-25650x+1570826\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-25650xz^2+1570826z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-410400x+100532880\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(84, 94)$ | $2.4620270109328603237875405251$ | $\infty$ |
Integral points
\( \left(84, 94\right) \), \( \left(84, -95\right) \)
Invariants
| Conductor: | $N$ | = | \( 6975 \) | = | $3^{2} \cdot 5^{2} \cdot 31$ |
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| Discriminant: | $\Delta$ | = | $14087689478325$ | = | $3^{9} \cdot 5^{2} \cdot 31^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{3792752640000}{28629151} \) | = | $2^{15} \cdot 3^{3} \cdot 5^{4} \cdot 19^{3} \cdot 31^{-5}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3529176691585451663441457881$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.26071880058511283536425197154$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1070685424336508$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.753681709627411$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.4620270109328603237875405251$ |
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| Real period: | $\Omega$ | ≈ | $0.70811742714096874475744809266$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.4868084650666935590419146859 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.486808465 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.708117 \cdot 2.462027 \cdot 2}{1^2} \\ & \approx 3.486808465\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 14400 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
| $31$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
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 |
|---|---|---|
| $5$ | 5Ns.2.1 | 5.30.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \), index $120$, genus $5$, and generators
$\left(\begin{array}{rr} 3 & 10 \\ 631 & 37 \end{array}\right),\left(\begin{array}{rr} 921 & 10 \\ 920 & 11 \end{array}\right),\left(\begin{array}{rr} 871 & 10 \\ 635 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 5 \\ 15 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 289 \\ 93 & 94 \end{array}\right)$.
The torsion field $K:=\Q(E[930])$ is a degree-$1028505600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/930\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 2325 = 3 \cdot 5^{2} \cdot 31 \) |
| $3$ | additive | $2$ | \( 775 = 5^{2} \cdot 31 \) |
| $5$ | additive | $10$ | \( 9 = 3^{2} \) |
| $31$ | nonsplit multiplicative | $32$ | \( 225 = 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 6975a consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 6975b1, its twist by $-3$.
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 |
|---|---|---|---|
| $3$ | 3.3.9300.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.8043570000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.31558444171875.2 | \(\Z/3\Z\) | not in database |
| $8$ | 8.4.56953125.1 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $16$ | 16.0.3243658447265625.1 | \(\Z/5\Z \oplus \Z/5\Z\) | 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 | ss | add | add | ss | ord | ss | ord | ord | ord | ord | nonsplit | ord | ss | ord | ss |
| $\lambda$-invariant(s) | 1,8 | - | - | 3,1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1,1 |
| $\mu$-invariant(s) | 0,0 | - | - | 0,0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.