Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2+178596x-19501428\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z+178596xz^2-19501428z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+231460389x-913330522314\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(13631056/45369, 74260573772/9663597)$ | $17.127063845997303892039700527$ | $\infty$ |
| $(103, -52)$ | $0$ | $2$ |
Integral points
\( \left(103, -52\right) \)
Invariants
| Conductor: | $N$ | = | \( 123981 \) | = | $3 \cdot 11 \cdot 13 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-530130101958954783$ | = | $-1 \cdot 3^{12} \cdot 11 \cdot 13 \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{26100282937247}{21962862207} \) | = | $3^{-12} \cdot 11^{-1} \cdot 13^{-1} \cdot 17^{-2} \cdot 29663^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0886571148910723981387491690$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.67205044286296435801398186006$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8949580184960225$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.083622326051458$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $17.127063845997303892039700527$ |
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| Real period: | $\Omega$ | ≈ | $0.16173147313534314487983774420$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.7699852662962197959632578841 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.769985266 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.161731 \cdot 17.127064 \cdot 4}{2^2} \\ & \approx 2.769985266\end{aligned}$$
Modular invariants
Modular form 123981.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1105920 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $2$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 58344 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 58337 & 8 \\ 58336 & 9 \end{array}\right),\left(\begin{array}{rr} 20860 & 30889 \\ 5831 & 37758 \end{array}\right),\left(\begin{array}{rr} 56203 & 56202 \\ 442 & 22747 \end{array}\right),\left(\begin{array}{rr} 44932 & 30889 \\ 42143 & 37758 \end{array}\right),\left(\begin{array}{rr} 3431 & 0 \\ 0 & 58343 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 58338 & 58339 \end{array}\right),\left(\begin{array}{rr} 41617 & 41616 \\ 29614 & 51919 \end{array}\right),\left(\begin{array}{rr} 38897 & 13736 \\ 45764 & 54945 \end{array}\right)$.
The torsion field $K:=\Q(E[58344])$ is a degree-$41625591585177600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/58344\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 41327 = 11 \cdot 13 \cdot 17^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 41327 = 11 \cdot 13 \cdot 17^{2} \) |
| $11$ | split multiplicative | $12$ | \( 11271 = 3 \cdot 13 \cdot 17^{2} \) |
| $13$ | split multiplicative | $14$ | \( 9537 = 3 \cdot 11 \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 429 = 3 \cdot 11 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 123981.d
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 7293.b4, its twist by $17$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-143}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2431}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{17}, \sqrt{-143})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | ord | nonsplit | ord | ss | split | split | add | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 9 | 1 | 1 | 1,1 | 2 | 2 | - | 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 |
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.