Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-91108x+16454788\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-91108xz^2+16454788z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7379775x+12017679750\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 84100 \) | = | $2^{2} \cdot 5^{2} \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $-68999505236000000$ | = | $-1 \cdot 2^{8} \cdot 5^{6} \cdot 29^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{35152}{29} \) | = | $-1 \cdot 2^{4} \cdot 13^{3} \cdot 29^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9290472481998955622997953718$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0214177433836885115370417253$ |
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| $abc$ quality: | $Q$ | ≈ | $0.714215699712365$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.123462920889655$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.31802074119210995011372749201$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 3\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.9081244471526597006823649521 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.908124447 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.318021 \cdot 1.000000 \cdot 6}{1^2} \\ & \approx 1.908124447\end{aligned}$$
Modular invariants
Modular form 84100.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 725760 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $29$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1740 = 2^{2} \cdot 3 \cdot 5 \cdot 29 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1735 & 6 \\ 1734 & 7 \end{array}\right),\left(\begin{array}{rr} 726 & 325 \\ 145 & 291 \end{array}\right),\left(\begin{array}{rr} 1043 & 0 \\ 0 & 1739 \end{array}\right),\left(\begin{array}{rr} 839 & 690 \\ 1125 & 329 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 871 & 1050 \\ 525 & 1411 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1740])$ is a degree-$94290739200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1740\Z)$.
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 | $2$ | \( 21025 = 5^{2} \cdot 29^{2} \) |
| $5$ | additive | $14$ | \( 3364 = 2^{2} \cdot 29^{2} \) |
| $29$ | additive | $450$ | \( 100 = 2^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 84100.e
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 116.b1, its twist by $145$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{145}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.116.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1560896.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.1107602046000.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.48778000.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.31801422347374898374550803368000000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.347849363699660931791255894016000000000.1 | \(\Z/6\Z\) | 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 | ord | add | ord | ord | ord | ord | ord | ord | add | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | 4 | - | 0 | 0 | 0 | 0 | 0 | 0 | - | 0 | 0 | 0,0 | 0 | 0 |
| $\mu$-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
All $p$-adic regulators are identically $1$ since the rank is $0$.