Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-74308035x+245321703117\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-74308035xz^2+245321703117z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-96303214035x+11447173928833902\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(94533/16, 6988797/64)$ | $6.9933253579123433700329425335$ | $\infty$ |
| $(18763/4, -18763/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 6762 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $254046278112105191473152$ | = | $2^{15} \cdot 3^{2} \cdot 7^{18} \cdot 23^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{385693937170561837203625}{2159357734550274048} \) | = | $2^{-15} \cdot 3^{-2} \cdot 5^{3} \cdot 7^{-12} \cdot 23^{-2} \cdot 59^{3} \cdot 443^{3} \cdot 557^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3333169734810116501162209035$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.3603618989533549975635445318$ |
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| $abc$ quality: | $Q$ | ≈ | $1.041884065687328$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.482054478560856$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.9933253579123433700329425335$ |
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| Real period: | $\Omega$ | ≈ | $0.098967053083637135273508700615$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.7684352077106262185104885907 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.768435208 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.098967 \cdot 6.993325 \cdot 16}{2^2} \\ & \approx 2.768435208\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1382400 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{15}$ | nonsplit multiplicative | 1 | 1 | 15 | 15 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{12}^{*}$ | additive | -1 | 2 | 18 | 12 |
| $23$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 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 | 8.6.0.6 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1114 & 1659 \\ 2457 & 2752 \end{array}\right),\left(\begin{array}{rr} 2857 & 2772 \\ 1134 & 1177 \end{array}\right),\left(\begin{array}{rr} 1275 & 2464 \\ 686 & 3557 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 3814 & 3855 \end{array}\right),\left(\begin{array}{rr} 1289 & 2772 \\ 3220 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1103 & 0 \\ 0 & 3863 \end{array}\right),\left(\begin{array}{rr} 3853 & 12 \\ 3852 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[3864])$ is a degree-$413653008384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3864\Z)$.
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 | $4$ | \( 49 = 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1127 = 7^{2} \cdot 23 \) |
| $5$ | good | $2$ | \( 3381 = 3 \cdot 7^{2} \cdot 23 \) |
| $7$ | additive | $32$ | \( 138 = 2 \cdot 3 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 6762.h
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 966.f1, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.829472.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{21})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.5667845604687.2 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.44033523122176.19 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.55729927701504.43 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.940615503487233299720214301429544042090496.2 | \(\Z/18\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | ss | add | ord | ord | ord | ord | split | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 5 | 1 | 1,1 | - | 1 | 1 | 1 | 1 | 2 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.