Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-2242x-38262\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-2242xz^2-38262z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-181629x-27348138\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(7411/25, 630416/125)$ | $5.5647102776408210154106113397$ | $\infty$ |
| $(-21, 0)$ | $0$ | $2$ |
Integral points
\( \left(-21, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 123008 \) | = | $2^{7} \cdot 31^{2}$ |
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| Discriminant: | $\Delta$ | = | $113600471168$ | = | $2^{7} \cdot 31^{6} $ |
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| j-invariant: | $j$ | = | \( 10976 \) | = | $2^{5} \cdot 7^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.86056386654034712357027604169$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2607655910288607632210248581$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8884240490394303$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.9658196230245366$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.5647102776408210154106113397$ |
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| Real period: | $\Omega$ | ≈ | $0.69782854776304401387115519308$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8832136917681795833521740161 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.883213692 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.697829 \cdot 5.564710 \cdot 4}{2^2} \\ & \approx 3.883213692\end{aligned}$$
Modular invariants
Modular form 123008.2.a.b
For more coefficients, see the Downloads section to the right.
| Modular degree: | 122880 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | -1 | 7 | 7 | 0 |
| $31$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 |
|---|---|---|
| $2$ | 2B | 32.48.0.18 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 992 = 2^{5} \cdot 31 \), index $96$, genus $0$, and generators
$\left(\begin{array}{rr} 985 & 8 \\ 984 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 927 & 0 \\ 0 & 991 \end{array}\right),\left(\begin{array}{rr} 962 & 899 \\ 713 & 32 \end{array}\right),\left(\begin{array}{rr} 63 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 8 \\ 20 & 23 \end{array}\right)$.
The torsion field $K:=\Q(E[992])$ is a degree-$3656908800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/992\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 961 = 31^{2} \) |
| $31$ | additive | $482$ | \( 128 = 2^{7} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 123008c
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 128a2, its twist by $-248$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.492032.2 | \(\Z/4\Z\) | not in database |
| $8$ | 8.4.61976445190144.12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.3873527824384.33 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | add | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | - | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - | 0 | 0 | 0 | 0 |
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.