Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-17805155x+28922059847\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-17805155xz^2+28922059847z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-284882475x+1850726947750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(4699, 219000\right) \) | $6.1667488966478319712638549885$ | $\infty$ |
| \( \left(\frac{9771}{4}, -\frac{9775}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([4699:219000:1]\) | $6.1667488966478319712638549885$ | $\infty$ |
| \([19542:-9775:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(18795, 1770800\right) \) | $6.1667488966478319712638549885$ | $\infty$ |
| \( \left(9770, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(4699, 219000\right) \), \( \left(4699, -223700\right) \)
\([4699:219000:1]\), \([4699:-223700:1]\)
\((18795,\pm 1770800)\)
Invariants
| Conductor: | $N$ | = | \( 72450 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 23$ |
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| Minimal Discriminant: | $\Delta$ | = | $7262753983257937500$ | = | $2^{2} \cdot 3^{22} \cdot 5^{6} \cdot 7 \cdot 23^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{54804145548726848737}{637608031452} \) | = | $2^{-2} \cdot 3^{-16} \cdot 7^{-1} \cdot 23^{-2} \cdot 193^{3} \cdot 19681^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7699996535909427444229938671$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4159745530398377114249915820$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0156853826442096$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.513405207165777$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.1667488966478319712638549885$ |
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| Real period: | $\Omega$ | ≈ | $0.21370045281381583517446027559$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $10.542696252821927045726188887 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.542696253 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.213700 \cdot 6.166749 \cdot 32}{2^2} \\ & \approx 10.542696253\end{aligned}$$
Modular invariants
Modular form 72450.2.a.dp
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4194304 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $4$ | $I_{16}^{*}$ | additive | -1 | 2 | 22 | 16 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $23$ | $2$ | $I_{2}$ | nonsplit 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 16.24.0.13 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 38640 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 15 & 2 \\ 38542 & 38627 \end{array}\right),\left(\begin{array}{rr} 7741 & 10320 \\ 18420 & 18361 \end{array}\right),\left(\begin{array}{rr} 23183 & 0 \\ 0 & 38639 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 25759 & 0 \\ 0 & 38639 \end{array}\right),\left(\begin{array}{rr} 5536 & 12885 \\ 10995 & 38626 \end{array}\right),\left(\begin{array}{rr} 38326 & 3225 \\ 26085 & 29626 \end{array}\right),\left(\begin{array}{rr} 17821 & 10320 \\ 7920 & 28021 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 38625 & 16 \\ 38624 & 17 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 38636 & 38637 \end{array}\right)$.
The torsion field $K:=\Q(E[38640])$ is a degree-$1588427552194560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/38640\Z)$.
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 | $4$ | \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \) |
| $3$ | additive | $8$ | \( 8050 = 2 \cdot 5^{2} \cdot 7 \cdot 23 \) |
| $5$ | additive | $14$ | \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 10350 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 72450.dp
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 966.g1, its twist by $-15$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{46})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{322})\) | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.24395696640000.46 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | split | add | add | nonsplit | ord | ord | ord | ord | nonsplit | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | - | - | 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 |
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.