Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-38x+90\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-38xz^2+90z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-608x+5776\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(0, 9)$ | $0.18129167403579927415788154138$ | $\infty$ |
Integral points
\( \left(0, 9\right) \), \( \left(0, -10\right) \), \( \left(4, 1\right) \), \( \left(4, -2\right) \)
Invariants
| Conductor: | $N$ | = | \( 361 \) | = | $19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-6859$ | = | $-1 \cdot 19^{3} $ |
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| j-invariant: | $j$ | = | \( -884736 \) | = | $-1 \cdot 2^{15} \cdot 3^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-19})/2]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.35014865989077482340427543809$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0862584046823849384065322961$ |
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| $abc$ quality: | $Q$ | ≈ | $1.3175706029138485$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8255696401648254$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.18129167403579927415788154138$ |
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| Real period: | $\Omega$ | ≈ | $4.1905500198059170003915208556$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.5194236564427329942747708344 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.519423656 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 4.190550 \cdot 0.181292 \cdot 2}{1^2} \\ & \approx 1.519423656\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 20 |
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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 not semistable. There is only one prime $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $19$ | $2$ | $III$ | additive | 1 | 2 | 3 | 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 |
|---|---|---|
| $19$ | 19B.1.6 | 19.360.7.9 |
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 |
|---|---|---|---|
| $19$ | additive | $110$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
19.
Its isogeny class 361.a
consists of 2 curves linked by isogenies of
degree 19.
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 |
|---|---|---|---|
| $3$ | 3.1.76.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.102889341747.1 | \(\Z/3\Z\) | not in database |
| $8$ | 8.0.5880735125.1 | \(\Z/5\Z\) | not in database |
| $9$ | \(\Q(\zeta_{19})^+\) | \(\Z/19\Z\) | not in database |
| $12$ | deg 12 | \(\Z/7\Z\) | not in database |
| $12$ | 12.2.338362575386312704.1 | \(\Z/4\Z\) | not in database |
| $16$ | 16.0.10586216645130957012009.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $16$ | 16.4.540360087662636962890625.1 | \(\Z/5\Z\) | not in database |
| $20$ | 20.0.35796303426870968668217777609.1 | \(\Z/11\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | ss | ord | ord | ord | ss | ord | add | ord | ss | ss | ss | ss | ord | ord |
| $\lambda$-invariant(s) | ? | 5,1 | 1 | 1 | 1 | 1,1 | 1 | - | 1 | 1,1 | 1,1 | 1,1 | 1,1 | 1 | 1 |
| $\mu$-invariant(s) | ? | 0,0 | 0 | 0 | 0 | 0,0 | 0 | - | 0 | 0,0 | 0,0 | 0,0 | 0,0 | 0 | 0 |
An entry ? indicates that the invariants have not yet been computed.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.