Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-115407x+15809053\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-115407xz^2+15809053z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1846515x+1009932878\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-394, 197)$ | $0$ | $2$ |
Integral points
\( \left(-394, 197\right) \)
Invariants
| Conductor: | $N$ | = | \( 149454 \) | = | $2 \cdot 3^{2} \cdot 19^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $-9200776496383728$ | = | $-1 \cdot 2^{4} \cdot 3^{12} \cdot 19^{6} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( -\frac{4956477625}{268272} \) | = | $-1 \cdot 2^{-4} \cdot 3^{-6} \cdot 5^{3} \cdot 11^{3} \cdot 23^{-1} \cdot 31^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8216558784698707298844204225$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.19986975544740434581771591191$ |
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| $abc$ quality: | $Q$ | ≈ | $0.950715771479093$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.917136537131244$ | |||
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: | $\Omega$ | ≈ | $0.40533384518689975482702974585$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.6213353807475990193081189834 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.621335381 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.405334 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 1.621335381\end{aligned}$$
Modular invariants
Modular form 149454.2.a.w
For more coefficients, see the Downloads section to the right.
| Modular degree: | 912384 |
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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 are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $19$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $23$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5244 = 2^{2} \cdot 3 \cdot 19 \cdot 23 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1407 & 3154 \\ 4598 & 4409 \end{array}\right),\left(\begin{array}{rr} 5233 & 12 \\ 5232 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 2659 & 1102 \\ 456 & 5243 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 4691 & 0 \\ 0 & 5243 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 5194 & 5235 \end{array}\right),\left(\begin{array}{rr} 970 & 3591 \\ 969 & 4960 \end{array}\right)$.
The torsion field $K:=\Q(E[5244])$ is a degree-$1578898759680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5244\Z)$.
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 | $4$ | \( 74727 = 3^{2} \cdot 19^{2} \cdot 23 \) |
| $3$ | additive | $2$ | \( 16606 = 2 \cdot 19^{2} \cdot 23 \) |
| $19$ | additive | $182$ | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 149454cb
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 138b1, its twist by $57$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-23}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{57}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.1195632.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-23}, \sqrt{57})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.276397836336.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.756224480215296.18 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.1429535879424.4 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.35962217868565460133724584426885730605312.1 | \(\Z/18\Z\) | 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 | 19 | 23 |
|---|---|---|---|---|
| Reduction type | nonsplit | add | add | split |
| $\lambda$-invariant(s) | 4 | - | - | 1 |
| $\mu$-invariant(s) | 0 | - | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.