Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-86x-2456\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-86xz^2-2456z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-110835x-114242994\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 38 \) | = | $2 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-2550136832$ | = | $-1 \cdot 2^{27} \cdot 19 $ |
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| j-invariant: | $j$ | = | \( -\frac{69173457625}{2550136832} \) | = | $-1 \cdot 2^{-27} \cdot 5^{3} \cdot 19^{-1} \cdot 821^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.48516961095724358095196192008$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.48516961095724358095196192008$ |
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| $abc$ quality: | $Q$ | ≈ | $1.054621266416818$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $8.007977586282983$ | |||
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: | $\Omega$ | ≈ | $0.63021074331409951207066938986$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.63021074331409951207066938986 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.630210743 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.630211 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 0.630210743\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{27}$ | nonsplit multiplicative | 1 | 1 | 27 | 27 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 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.1.2 | 27.72.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4104 = 2^{3} \cdot 3^{3} \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 28 & 27 \\ 1521 & 3592 \end{array}\right),\left(\begin{array}{rr} 937 & 75 \\ 1653 & 2144 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 2386 & 1447 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 688 & 9 \\ 791 & 664 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right),\left(\begin{array}{rr} 3079 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4051 & 54 \\ 4050 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[4104])$ is a degree-$45954293760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4104\Z)$.
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 | $4$ | \( 19 \) |
| $3$ | good | $2$ | \( 19 \) |
| $19$ | split multiplicative | $20$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 38a
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-1444.2-b4 |
| $3$ | 3.1.152.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.1083.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.3511808.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.3518667.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.2565108243.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.7105563.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.623808.1 | \(\Z/6\Z\) | not in database |
| $9$ | 9.1.12356882919936.1 | \(\Z/6\Z\) | not in database |
| $12$ | 12.2.119973433931988992.10 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.8990607867641856.3 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.16877848680315122776257224907.3 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.4122698998419163225428590592.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.1597218061967362443766361801733439488.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.12256029818428054141438155030528.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.1485393179874874809517382565888.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 19 |
|---|---|---|---|
| Reduction type | nonsplit | ord | split |
| $\lambda$-invariant(s) | 1 | 0 | 1 |
| $\mu$-invariant(s) | 0 | 2 | 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 $0$.