Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-722861x-240965848\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-722861xz^2-240965848z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-936827235x-11239692110946\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 90354 \) | = | $2 \cdot 3 \cdot 11 \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $-897621149408419392$ | = | $-1 \cdot 2^{6} \cdot 3 \cdot 11^{3} \cdot 37^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{11892507625}{255552} \) | = | $-1 \cdot 2^{-6} \cdot 3^{-1} \cdot 5^{3} \cdot 11^{-3} \cdot 37 \cdot 137^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2347839703569609897712474765$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.17249463807252197314081630419$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8954757725847072$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.567587094333869$ | |||
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: | $\Omega$ | ≈ | $0.081762367230477208450411598990$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 2\cdot1\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.49057420338286325070246959394 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.490574203 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.081762 \cdot 1.000000 \cdot 6}{1^2} \\ & \approx 0.490574203\end{aligned}$$
Modular invariants
Modular form 90354.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1342656 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $37$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
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.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 132 = 2^{2} \cdot 3 \cdot 11 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 14 & 123 \\ 1 & 34 \end{array}\right),\left(\begin{array}{rr} 67 & 6 \\ 69 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 129 & 130 \\ 122 & 125 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 6 \\ 39 & 19 \end{array}\right),\left(\begin{array}{rr} 127 & 6 \\ 126 & 7 \end{array}\right)$.
The torsion field $K:=\Q(E[132])$ is a degree-$3801600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/132\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 45177 = 3 \cdot 11 \cdot 37^{2} \) |
| $3$ | split multiplicative | $4$ | \( 1369 = 37^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 8214 = 2 \cdot 3 \cdot 37^{2} \) |
| $37$ | additive | $506$ | \( 66 = 2 \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 90354h
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 90354u2, its twist by $37$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.180708.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.332667.2 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.4310510326848.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.332001998667.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.97966143792.4 | \(\Z/6\Z\) | not in database |
| $9$ | 9.1.9408241339733027664576.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.52269910320587460220418437636471898706029458010112.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.265545015319984545823579283314685234295779328.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.22620186585017563551435777669878146998251666276352.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 11 | 37 |
|---|---|---|---|---|
| Reduction type | nonsplit | split | nonsplit | add |
| $\lambda$-invariant(s) | 2 | 3 | 0 | - |
| $\mu$-invariant(s) | 0 | 1 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
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 $0$.