Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-93701832x-349092134751\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-93701832xz^2-349092134751z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1499229307x-22343395853354\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 46090 \) | = | $2 \cdot 5 \cdot 11 \cdot 419$ |
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| Discriminant: | $\Delta$ | = | $249396782289047639290$ | = | $2 \cdot 5 \cdot 11 \cdot 419^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{90984613355465878035683930961}{249396782289047639290} \) | = | $2^{-1} \cdot 3^{3} \cdot 5^{-1} \cdot 11^{-1} \cdot 37^{3} \cdot 167^{3} \cdot 419^{-7} \cdot 242633^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1470810512577897182177518590$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.1470810512577897182177518590$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0059525187140597$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.209564778138736$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.048524671718521780245935584900$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.3777089142075672320508436601 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $49$ = $7^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.377708914 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{49 \cdot 0.048525 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 2.377708914\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9938768 |
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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 semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $419$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
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 |
|---|---|---|
| $7$ | 7B.1.3 | 7.48.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1290520 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \cdot 419 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 1032417 & 14 \\ 774319 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 322631 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1290507 & 14 \\ 1290506 & 15 \end{array}\right),\left(\begin{array}{rr} 1055881 & 14 \\ 938567 & 99 \end{array}\right),\left(\begin{array}{rr} 645261 & 14 \\ 645267 & 99 \end{array}\right),\left(\begin{array}{rr} 1114961 & 14 \\ 61607 & 99 \end{array}\right),\left(\begin{array}{rr} 322633 & 368728 \\ 645246 & 875673 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right)$.
The torsion field $K:=\Q(E[1290520])$ is a degree-$6284077839687352320000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1290520\Z)$.
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 | $4$ | \( 23045 = 5 \cdot 11 \cdot 419 \) |
| $5$ | split multiplicative | $6$ | \( 9218 = 2 \cdot 11 \cdot 419 \) |
| $7$ | good | $2$ | \( 110 = 2 \cdot 5 \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 4190 = 2 \cdot 5 \cdot 419 \) |
| $419$ | nonsplit multiplicative | $420$ | \( 110 = 2 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 46090p
consists of 2 curves linked by isogenies of
degree 7.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $3$ | 3.3.184360.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.6266140065856000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | \(\Q(\zeta_{7})\) | \(\Z/7\Z\) | not in database |
| $7$ | 7.1.1458956660623000000.12 | \(\Z/7\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.0.186410682672938925533420597960421634048000000.1 | \(\Z/14\Z\) | not in database |
| $21$ | 21.3.12689308569287721859126024973385805726358306194789082389381120000000000000000000.1 | \(\Z/14\Z\) | 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 | 419 |
|---|---|---|---|---|---|---|
| Reduction type | split | ss | split | ord | split | nonsplit |
| $\lambda$-invariant(s) | 1 | 0,0 | 1 | 4 | 1 | 0 |
| $\mu$-invariant(s) | 0 | 0,0 | 0 | 1 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.