Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-68480997x-218124502739\)
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(homogenize, simplify) |
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\(y^2z=x^3-68480997xz^2-218124502739z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-68480997x-218124502739\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(135057, 49538125\right) \) | $5.2446405695957562134012793773$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([135057:49538125:1]\) | $5.2446405695957562134012793773$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(135057, 49538125\right) \) | $5.2446405695957562134012793773$ | $\infty$ |
Integral points
\((135057,\pm 49538125)\)
\([135057:\pm 49538125:1]\)
\((135057,\pm 49538125)\)
Invariants
| Conductor: | $N$ | = | \( 91260 \) | = | $2^{2} \cdot 3^{3} \cdot 5 \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-116318102497593750000$ | = | $-1 \cdot 2^{4} \cdot 3^{3} \cdot 5^{9} \cdot 13^{10} $ |
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| j-invariant: | $j$ | = | \( -\frac{17032957979647640832}{55783203125} \) | = | $-1 \cdot 2^{8} \cdot 3^{3} \cdot 5^{-9} \cdot 13^{-4} \cdot 19^{3} \cdot 7109^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0732249675822215815592455289$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2850481564977773542112797917$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0493635705233648$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.7558109058812335$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.2446405695957562134012793773$ |
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| Real period: | $\Omega$ | ≈ | $0.026240792982586007671835382194$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 54 $ = $ 3\cdot1\cdot3^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.4316704825610468988958909568 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.431670483 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.026241 \cdot 5.244641 \cdot 54}{1^2} \\ & \approx 7.431670483\end{aligned}$$
Modular invariants
Modular form 91260.2.a.t
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6531840 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $1$ | $II$ | additive | 1 | 3 | 3 | 0 |
| $5$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $13$ | $2$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 385 & 6 \\ 384 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 157 & 156 \\ 351 & 79 \end{array}\right),\left(\begin{array}{rr} 248 & 117 \\ 91 & 196 \end{array}\right),\left(\begin{array}{rr} 89 & 0 \\ 0 & 389 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[390])$ is a degree-$226437120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/390\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 22815 = 3^{3} \cdot 5 \cdot 13^{2} \) |
| $3$ | additive | $2$ | \( 676 = 2^{2} \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 18252 = 2^{2} \cdot 3^{3} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 91260.t
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 7020.h1, its twist by $13$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.135.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.273375.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.53466192.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.120120975.4 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.5027504430864820649656553689331920896.3 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.11422347092018337581176419648000000.3 | \(\Z/6\Z\) | 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 | add | add | split | ord | ss | add | ord | ord | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | - | 2 | 1 | 1,1 | - | 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 | 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.