Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-94975806x+356224915116\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-94975806xz^2+356224915116z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-123088645251x+16621875969327486\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-7089, 824106)$ | $4.8205541013472605074200874679$ | $\infty$ |
Integral points
\( \left(-7089, 824106\right) \), \( \left(-7089, -817017\right) \)
Invariants
| Conductor: | $N$ | = | \( 122694 \) | = | $2 \cdot 3 \cdot 11^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1246395839798695456728$ | = | $-1 \cdot 2^{3} \cdot 3^{4} \cdot 11^{9} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{49258558427}{648} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-4} \cdot 13 \cdot 1559^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1919077239099000615057538382$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.31647996899656900390969547298$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9763153716075734$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.69415967162148$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.8205541013472605074200874679$ |
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| Real period: | $\Omega$ | ≈ | $0.13967163097138231223698496894$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.6931786140838322413446952077 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.693178614 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.139672 \cdot 4.820554 \cdot 4}{1^2} \\ & \approx 2.693178614\end{aligned}$$
Modular invariants
Modular form 122694.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 14826240 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $13$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 88 = 2^{3} \cdot 11 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 87 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 87 & 2 \\ 86 & 3 \end{array}\right),\left(\begin{array}{rr} 23 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 45 & 2 \\ 45 & 3 \end{array}\right),\left(\begin{array}{rr} 57 & 2 \\ 57 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[88])$ is a degree-$10137600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/88\Z)$.
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 | $4$ | \( 1859 = 11 \cdot 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 20449 = 11^{2} \cdot 13^{2} \) |
| $11$ | additive | $42$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $74$ | \( 726 = 2 \cdot 3 \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 122694h consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 122694g1, its twist by $-143$.
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 |
|---|---|---|---|
| $3$ | 3.1.14872.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.19463521792.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | ord | ord | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 5 | 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 |
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.