Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-523x-4642\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-523xz^2-4642z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-677187x-214533954\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 930 \) | = | $2 \cdot 3 \cdot 5 \cdot 31$ |
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| Minimal Discriminant: | $\Delta$ | = | $-3574920$ | = | $-1 \cdot 2^{3} \cdot 3 \cdot 5 \cdot 31^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{15777367606441}{3574920} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{3} \cdot 31^{-3} \cdot 3583^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.24960489030348196138877929831$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.24960489030348196138877929831$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9449610246978523$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.446110954118944$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $0.49927262319524691058246737932$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3 $ = $ 1\cdot1\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.4978178695857407317474021380 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.497817870 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.499273 \cdot 1.000000 \cdot 3}{1^2} \\ & \approx 1.497817870\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 360 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $31$ | $3$ | $I_{3}$ | 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.1.2 | 3.8.0.2 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1801 & 6 \\ 1683 & 19 \end{array}\right),\left(\begin{array}{rr} 1861 & 6 \\ 1863 & 19 \end{array}\right),\left(\begin{array}{rr} 3715 & 6 \\ 3714 & 7 \end{array}\right),\left(\begin{array}{rr} 2791 & 6 \\ 933 & 19 \end{array}\right),\left(\begin{array}{rr} 2977 & 6 \\ 1491 & 19 \end{array}\right),\left(\begin{array}{rr} 2018 & 1707 \\ 1 & 466 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3720])$ is a degree-$1974730752000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3720\Z)$.
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 | $4$ | \( 465 = 3 \cdot 5 \cdot 31 \) |
| $3$ | split multiplicative | $4$ | \( 5 \) |
| $5$ | split multiplicative | $6$ | \( 186 = 2 \cdot 3 \cdot 31 \) |
| $31$ | split multiplicative | $32$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 930i
consists of 2 curves linked by isogenies of
degree 3.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
| $3$ | 3.1.3720.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.6075.2 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.51478848000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.110716875.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.41515200.1 | \(\Z/6\Z\) | not in database |
| $9$ | 9.1.51296194612440000000.1 | \(\Z/6\Z\) | 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 |
| $18$ | 18.0.778429354862626818016425111978667200000000.3 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.7893898745151955445408260800000000000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.602027872043064075965843193581568000000000000000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 5 | 31 |
|---|---|---|---|---|
| Reduction type | nonsplit | split | split | split |
| $\lambda$-invariant(s) | 2 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 1 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.