This is a model for the modular curve $X_0(19)$.
Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2-9x-15\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z-9xz^2-15z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-12096x-544752\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5, 9)$ | $0$ | $3$ |
Integral points
\( \left(5, 9\right) \), \( \left(5, -10\right) \)
Invariants
| Conductor: | $N$ | = | \( 19 \) | = | $19$ |
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| Discriminant: | $\Delta$ | = | $-6859$ | = | $-1 \cdot 19^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{89915392}{6859} \) | = | $-1 \cdot 2^{18} \cdot 7^{3} \cdot 19^{-3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.51586698734273704217752773488$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.51586698734273704217752773488$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0331037033479094$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.262043087240102$ | |||
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: | $\Omega$ | ≈ | $1.3597597334883108107365175612$ |
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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)$ | ≈ | $0.45325324449610360357883918707 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.453253244 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.359760 \cdot 1.000000 \cdot 3}{3^2} \\ & \approx 0.453253244\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $19$ | $3$ | $I_{3}$ | 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 \( 1026 = 2 \cdot 3^{3} \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 43 & 30 \\ 840 & 85 \end{array}\right),\left(\begin{array}{rr} 892 & 27 \\ 15 & 676 \end{array}\right),\left(\begin{array}{rr} 935 & 790 \\ 714 & 553 \end{array}\right),\left(\begin{array}{rr} 973 & 54 \\ 972 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 27 \\ 27 & 730 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19 & 54 \\ 792 & 469 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1026])$ is a degree-$179508960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1026\Z)$.
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 | $2$ | \( 1 \) |
| $19$ | split multiplicative | $20$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 19a
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z \oplus \Z/3\Z\) | 2.0.3.1-361.2-a4 |
| $3$ | 3.1.76.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.155952.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $9$ | 9.3.1270238787.1 | \(\Z/9\Z\) | not in database |
| $12$ | 12.2.937292452593664.2 | \(\Z/12\Z\) | not in database |
| $12$ | 12.0.8779890495744.1 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.358752951832597688547.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.16877848680315122776257224907.2 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.43564677551979246963.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 19 |
|---|---|---|---|
| Reduction type | ss | ord | split |
| $\lambda$-invariant(s) | 0,3 | 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 $0$.