Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-1810330x-903698732\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-1810330xz^2-903698732z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2346188355x-42127775218242\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(18921, 2586515)$ | $7.1798421184844526705709922157$ | $\infty$ |
| $(-652, 326)$ | $0$ | $2$ |
Integral points
\( \left(-652, 326\right) \), \( \left(18921, 2586515\right) \), \( \left(18921, -2605436\right) \)
Invariants
| Conductor: | $N$ | = | \( 119658 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $27498031011386413632$ | = | $2^{6} \cdot 3^{11} \cdot 7^{6} \cdot 11 \cdot 37^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{5577108481460841625}{233729407061568} \) | = | $2^{-6} \cdot 3^{-11} \cdot 5^{3} \cdot 11^{-1} \cdot 37^{-4} \cdot 354677^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4948473823279300749977701535$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5218923078002734224450937818$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9869944369461394$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.690285661668411$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.1798421184844526705709922157$ |
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| Real period: | $\Omega$ | ≈ | $0.13049152675510938838513471914$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot2\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.7476342396067009081928363476 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.747634240 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.130492 \cdot 7.179842 \cdot 16}{2^2} \\ & \approx 3.747634240\end{aligned}$$
Modular invariants
Modular form 119658.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4561920 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $1$ | $I_{11}$ | nonsplit multiplicative | 1 | 1 | 11 | 11 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $37$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 | 8.6.0.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 178 & 1 \\ 175 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 26 & 1 \\ 119 & 0 \end{array}\right),\left(\begin{array}{rr} 261 & 4 \\ 260 & 5 \end{array}\right),\left(\begin{array}{rr} 169 & 100 \\ 32 & 231 \end{array}\right),\left(\begin{array}{rr} 133 & 4 \\ 2 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$81100800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 | $4$ | \( 1617 = 3 \cdot 7^{2} \cdot 11 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 19943 = 7^{2} \cdot 11 \cdot 37 \) |
| $7$ | additive | $26$ | \( 2442 = 2 \cdot 3 \cdot 11 \cdot 37 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 3626 = 2 \cdot 7^{2} \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 119658i
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2442f1, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.103488.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.11662935330816.24 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.3175234143814656.14 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | nonsplit | nonsplit | ss | add | nonsplit | ord | ord | ord | ord | ord | ord | split | ord | ord | ord |
| $\lambda$-invariant(s) | 7 | 1 | 1,1 | - | 3 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 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.