Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2-23x-50\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z-23xz^2-50z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-30240x-1959984\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(8, 18)$ | $0$ | $3$ |
Integral points
\( \left(8, 18\right) \), \( \left(8, -19\right) \)
Invariants
| Conductor: | $N$ | = | \( 37 \) | = | $37$ |
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| Discriminant: | $\Delta$ | = | $50653$ | = | $37^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{1404928000}{50653} \) | = | $2^{15} \cdot 5^{3} \cdot 7^{3} \cdot 37^{-3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.32722458481035149069083761293$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.32722458481035149069083761293$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9727427413985318$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.8332126075508315$ | |||
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: | $\Omega$ | ≈ | $2.1770431858084583470086166231$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3 $ = $ 3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.72568106193615278233620554103 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.725681062 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.177043 \cdot 1.000000 \cdot 3}{3^2} \\ & \approx 0.725681062\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2 |
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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 semistable. There is only one prime $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $37$ | $3$ | $I_{3}$ | 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 |
|---|---|---|
| $3$ | 3Cs.1.1 | 9.72.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1998 = 2 \cdot 3^{3} \cdot 37 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 19 & 54 \\ 576 & 901 \end{array}\right),\left(\begin{array}{rr} 1 & 27 \\ 27 & 730 \end{array}\right),\left(\begin{array}{rr} 43 & 30 \\ 1704 & 1003 \end{array}\right),\left(\begin{array}{rr} 1799 & 844 \\ 984 & 1039 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 352 & 27 \\ 717 & 1648 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 1945 & 54 \\ 1944 & 55 \end{array}\right)$.
The torsion field $K:=\Q(E[1998])$ is a degree-$2656732608$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1998\Z)$.
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 |
|---|---|---|---|
| $3$ | good | $2$ | \( 1 \) |
| $37$ | split multiplicative | $38$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 37.b
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z \oplus \Z/3\Z\) | 2.0.3.1-1369.2-b3 |
| $3$ | 3.3.148.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.6.810448.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.591408.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $9$ | 9.3.69274613043.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | 12.0.478826125353216.1 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.19565196851635126921683.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.50198942259523899975028947826347.3 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.129572244330949414435923.1 | \(\Z/3\Z \oplus \Z/9\Z\) | 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 | 37 |
|---|---|---|---|
| Reduction type | ss | ord | split |
| $\lambda$-invariant(s) | 0,5 | 0 | 1 |
| $\mu$-invariant(s) | 0,0 | 1 | 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 $0$.
Additional information
This is the last elliptic curve over $\Q$ of prime conductor $N$ and discriminant $\Delta = \pm N^e$ with $e > 2$ (and, as it happens, the only one with $\Delta > 0$). The others are the modular curves $X_0(N)$ for N=11 [11.a2], N=17 [17.a3], and N=19 [19.a2] (with $e=5,4,3$ respectively -- in each case, as here with $(N,e)=(37,3)$, the exponent $e$ is the numerator of $(N-1)/12$).