Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-2300486x-1342427691\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-2300486xz^2-1342427691z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-36807771x-85952179978\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{21879}{25}, \frac{59191}{125}\right) \) | $5.6343051536324402166349861135$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-109395:59191:125]\) | $5.6343051536324402166349861135$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{87541}{25}, \frac{36448}{125}\right) \) | $5.6343051536324402166349861135$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 69678 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 79$ |
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| Minimal Discriminant: | $\Delta$ | = | $1776162847850496$ | = | $2^{18} \cdot 3^{6} \cdot 7^{6} \cdot 79 $ |
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| j-invariant: | $j$ | = | \( \frac{15698803397448457}{20709376} \) | = | $2^{-18} \cdot 11^{3} \cdot 13^{3} \cdot 17^{3} \cdot 79^{-1} \cdot 103^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2021192882434968127235193026$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.67985806938178531447322031242$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0014551545932482$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.982181631578665$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.6343051536324402166349861135$ |
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| Real period: | $\Omega$ | ≈ | $0.12258708135031076749215768036$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 18 $ = $ ( 2 \cdot 3^{2} )\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.432474435974672802929222041 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.432474436 \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 0.122587 \cdot 5.634305 \cdot 18}{1^2} \\ & \approx 12.432474436\end{aligned}$$
Modular invariants
Modular form 69678.2.a.bs
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1360800 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $18$ | $I_{18}$ | split multiplicative | -1 | 1 | 18 | 18 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $79$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3B | 9.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 19908 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 79 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 9953 & 5670 \\ 0 & 1105 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 17063 & 0 \\ 0 & 19907 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 9955 & 14238 \\ 7119 & 8695 \end{array}\right),\left(\begin{array}{rr} 16927 & 14238 \\ 18081 & 16717 \end{array}\right),\left(\begin{array}{rr} 19891 & 18 \\ 19890 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right)$.
The torsion field $K:=\Q(E[19908])$ is a degree-$200923996815360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19908\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 34839 = 3^{2} \cdot 7^{2} \cdot 79 \) |
| $3$ | additive | $6$ | \( 3871 = 7^{2} \cdot 79 \) |
| $7$ | additive | $26$ | \( 1422 = 2 \cdot 3^{2} \cdot 79 \) |
| $79$ | split multiplicative | $80$ | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 69678bs
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 158d2, its twist by $21$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.3.316.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.31554496.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.29218052711421.2 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.87654158134263.3 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.34250608.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.637628815438662947438358670359283996502376861696.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.17215978016843899580835684099700667905564175265792.1 | \(\Z/18\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 79 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | add | ord | add | ss | ord | ss | ord | ord | ss | ord | ord | ord | ord | ord | split |
| $\lambda$-invariant(s) | 2 | - | 1 | - | 1,1 | 1 | 5,1 | 3 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 2 |
| $\mu$-invariant(s) | 0 | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.